*Let one take* a = 1 *and Y*∗(*ω*) = *Y* ∗ (*ω*)*. Then, from Theorem 1 and (5), we achieve classical* H*–*H *type inequality.*

**Example 1.** *Let* a = 1 2 , *ω* ∈ [2, 3]*, and the I*·*V-F Y* : [*s*, *t*] = [2, 3] → X + *I* , *provided by Y*(*ω*) = [1, 2] 2 − *ω* 1 2 *. Since the left and right endpoint functions, Y*∗(*ω*) = 2 − *ω* 1 2 , *Y* ∗ (*ω*) = 2 2 − *ω* 1 2 *, are left and right convex functions, then Y*(*ω*) *is a left and right convex I*·*V-F. We clearly see that Y* ∈ *L* [*s*, *t*], X + *I , and*

$$\begin{array}{l} \mathcal{Y}\_{\*}\left(\frac{s+t}{2}\right) = \mathcal{Y}\_{\*}\left(\frac{5}{2}\right) = \frac{4-\sqrt{10}}{2} \\ \mathcal{Y}^{\*}\left(\frac{s+t}{2}\right) = \mathcal{Y}^{\*}\left(\frac{5}{2}\right) = 4-\sqrt{10} \\ \frac{\mathcal{Y}\_{\*}(s)+\mathcal{Y}\_{\*}(t)}{2} = \frac{4-\sqrt{2}-\sqrt{3}}{2} \\ \frac{\mathcal{Y}^{\*}(s)+\mathcal{Y}^{\*}(t)}{2} = 4-\sqrt{2}-\sqrt{3} \end{array}$$

*Note that*

Γ(a+1) 2(*t*−*s*) a - I a *s*+ *Y*∗(*t*) + I a *t*− *Y*∗(*s*) = Γ( 3 2 ) 2 √ 1 *π* R 3 2 (3 − *ω*) −1 2 . 2 − *ω* 1 2 *dω* + Γ( 3 2 ) 2 √ 1 *π* R 3 2 (*ω* − 2) −1 2 . 2 − *ω* 1 2 *dω* = 1 4 h 7393 10,000 + 9501 10,000 i = 8447 20,000 . Γ(a+1) 2(*t*−*s*) a - I a *s*+ *Y* ∗ (*t*) + I a *t*− *Y* ∗ (*s*) = Γ( 3 2 ) 2 √ 1 *π* R 3 2 (3 − *ω*) −1 2 . 2 2 − *ω* 1 2 *dω* + Γ( 3 2 ) 2 √ 1 *π* R 3 2 (*ω* − 2) −1 2 . 2 2 − *ω* 1 2 *dω* = 1 2 h 7393 10,000 + 9501 10,000 i = 8447 10,000

*Therefore*,

$$\left[\frac{4-\sqrt{10}}{2}, 4-\sqrt{10}\right] \le\_p \left[\frac{8447}{20,000}, \frac{8447}{10,000}\right] \le\_p \left[\frac{4-\sqrt{2}-\sqrt{3}}{2}, 4-\sqrt{2}-\sqrt{3}\right]$$

*and Theorem 3 is verified.*

The upcoming two results acquire the fractional inequalities for the product of left and right convex *I*·*V-Fs*.

**Theorem 4.** *Let Y*, G : [*s*, *t*] → X + *I be two left and right convex I*·*V-Fs on* [*s*, *t*], *provided by Y*(*ω*) = [*Y*∗(*ω*), *Y* ∗ (*ω*)] *and* G(*ω*) = [G∗(*ω*), G∗ (*ω*)] *for all ω* ∈ [*s*, *t*]*. If Y* × G ∈ *L* [*s*, *t*], X + *I , then*

$$\begin{aligned} &\frac{\Gamma(\mathfrak{a})}{2(t-s)^{\mathfrak{a}}} \Big[ \mathcal{Z}\_{s+}^{\mathfrak{a}} \, \, \, Y(t) \times \mathfrak{G}(t) + \mathcal{Z}\_{t-}^{\mathfrak{a}} \, \, Y(s) \times \mathfrak{G}(s) \Big] \\ &\leq\_{p} \left( \frac{1}{2} - \frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)} \right) \varphi(s,t) + \left( \frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)} \right) \nabla(s,t) \, \, \, \end{aligned}$$

where *ϕ*(*s*, *t*) = *Y*(*s*) × G(*s*) + *Y*(*t*) × G(*t*), ∇(*s*, *t*) = *Y*(*s*) × G(*t*) + *Y*(*t*) × G(*s*), and *ϕ*(*s*, *t*) = [*ϕ*∗(*s*, *t*), *ϕ* ∗ (*s*, *t*)] and ∇(*s*, *t*) = [∇∗(*s*, *t*), ∇∗ (*s*, *t*)].

**Proof.** Since *Y*, G are both left and right convex *I*·*V-Fs,* then we have

$$\begin{aligned} \mathcal{Y}\_\*(\nu s + (1 - \nu)t) &\leq \nu \mathcal{Y}\_\*(s) + (1 - \nu)\mathcal{Y}\_\*(t), \\ \mathcal{Y}^\*(\nu s + (1 - \nu)t) &\leq \nu \mathcal{Y}^\*(s) + (1 - \nu)\mathcal{Y}^\*(t). \end{aligned}$$

and

$$
\mathfrak{G}\_\*(\nu s + (1 - \nu)t) \le \nu \mathfrak{G}\_\*(s) + (1 - \nu)\mathfrak{G}\_\*(t),
$$

$$
\mathfrak{G}^\*(\nu s + (1 - \nu)t) \le \nu \mathfrak{G}^\*(s) + (1 - \nu)\mathfrak{G}^\*(t).
$$

From the definition of left and right convex *I*·*V-Fs,* it follows that 0 ≤*p Y*(*ω*) and 0 ≤*p* G(*ω*), so

$$\begin{array}{c} Y\_\*(\nu s + (1 - \nu)t) \times \mathfrak{G}\_\*(\nu s + (1 - \nu)t) \\ \leq & (\nu Y\_\*(s) + (1 - \nu)Y\_\*(t))(\nu \mathfrak{G}\_\*(s) + (1 - \nu)\mathfrak{G}\_\*(t)) \\ = & \nu^2 Y\_\*(s) \times \mathfrak{G}\_\*(s) + (1 - \nu)^2 Y\_\*(t) \times \mathfrak{G}\_\*(t) \\ \qquad + \nu (1 - \nu) Y\_\*(s) \times \mathfrak{G}\_\*(t) + \nu (1 - \nu) Y\_\*(t) \times \mathfrak{G}\_\*(s) \\ Y^\*(\nu s + (1 - \nu)t) \times \mathfrak{G}^\*(\nu s + (1 - \nu)t) \\ \leq & (\nu Y^\*(s) + (1 - \nu)Y^\*(t))(\nu \mathfrak{G}^\*(s) + (1 - \nu)\mathfrak{G}^\*(t)) \\ = & \nu^2 Y^\*(s) \times \mathfrak{G}^\*(s) + (1 - \nu)^2 Y^\*(t) \times \mathfrak{G}^\*(t) \\ + \nu (1 - \nu) Y^\*(s) \times \mathfrak{G}^\*(t) + \nu (1 - \nu) Y^\*(t) \times \mathfrak{G}^\*(s), \end{array} \tag{9}$$

Analogously, we have

$$\begin{array}{c} Y\_{\*}((1-\nu)s+\nu t)\mathfrak{G}\_{\*}((1-\nu)s+\nu t) \\ \leq & (1-\nu)^{2}Y\_{\*}(s)\times\mathfrak{G}\_{\*}(s)+\nu^{2}Y\_{\*}(t)\times\mathfrak{G}\_{\*}(t) \\ +\nu(1-\nu)Y\_{\*}(s)\times\mathfrak{G}\_{\*}(t)+\nu(1-\nu)Y\_{\*}(t)\times\mathfrak{G}\_{\*}(s) \\ Y^{\*}((1-\nu)s+\nu t)\times\mathfrak{G}^{\*}((1-\nu)s+\nu t) \\ \leq & (1-\nu)^{2}Y^{\*}(s)\times\mathfrak{G}^{\*}(s)+\nu^{2}Y^{\*}(t)\times\mathfrak{G}^{\*}(t) \\ +\nu(1-\nu)Y^{\*}(s)\times\mathfrak{G}^{\*}(t)+\nu(1-\nu)Y^{\*}(t)\times\mathfrak{G}^{\*}(s). \end{array} \tag{10}$$

Adding (9) and (10), we have

$$\begin{array}{l} Y\_{\*} (\upsilon s + (1 - \upsilon)t) \times \mathfrak{G}\_{\*} (\upsilon s + (1 - \upsilon)t) \\ \qquad \qquad + Y\_{\*} ((1 - \upsilon)s + \upsilon t) \times \mathfrak{G}\_{\*} ((1 - \upsilon)s + \upsilon t) \\ \qquad \qquad \leq \left[ \upsilon^{2} + (1 - \upsilon)^{2} \right] [Y\_{\*} (s) \times \mathfrak{G}\_{\*} (s) + Y\_{\*} (t) \times \mathfrak{G}\_{\*} (t)] \\ \qquad \qquad + 2\upsilon (1 - \upsilon) [Y\_{\*} (t) \times \mathfrak{G}\_{\*} (s) + Y\_{\*} (s) \times \mathfrak{G}\_{\*} (t)] \\ Y^{\*} (\upsilon s + (1 - \upsilon)t) \times \mathfrak{G}^{\*} (\upsilon s + (1 - \upsilon)t) \\ \qquad \qquad + Y^{\*} ((1 - \upsilon)s + \upsilon t) \times \mathfrak{G}^{\*} ((1 - \upsilon)s + \upsilon t) \\ \qquad \leq \left[ \upsilon^{2} + (1 - \upsilon)^{2} \right] [Y^{\*} (s) \times \mathfrak{G}^{\*} (s) + Y^{\*} (t) \times \mathfrak{G}^{\*} (t)] \\ \qquad \qquad + 2\upsilon (1 - \upsilon) [Y^{\*} (t) \times \mathfrak{G}^{\*} (s) + Y^{\*} (s) \times \mathfrak{G}^{\*} (t)]. \end{array} \tag{11}$$

Taking the multiplication of (11) by *ν* a−1 and integrating the obtained result, with respect to *ν* over (0, 1), we have

$$\begin{split} \int\_{0}^{1} \boldsymbol{\nu}^{\mathsf{a}-1} Y\_{\ast} (\boldsymbol{\nu}s + (1-\boldsymbol{\nu})t) &\times \mathfrak{G}\_{\ast} (\boldsymbol{\nu}s + (1-\boldsymbol{\nu})t) \\ &+ \boldsymbol{\nu}^{\mathsf{a}-1} Y\_{\ast} ((1-\boldsymbol{\nu})s + \boldsymbol{\nu}t) \times \mathfrak{G}\_{\ast} ((1-\boldsymbol{\nu})s + \boldsymbol{\nu}t) d\boldsymbol{\nu} \\ &\leq q\_{\ast}(s,t) \int\_{0}^{1} \boldsymbol{\nu}^{\mathsf{a}-1} \left[\boldsymbol{\nu}^{2} + (1-\boldsymbol{\nu})^{2}\right] d\boldsymbol{\nu} + 2\nabla\_{\ast}(s,t) \int\_{0}^{1} \boldsymbol{\nu}^{\mathsf{a}-1} \boldsymbol{\nu} (1-\boldsymbol{\nu}) d\boldsymbol{\nu} \\ \int\_{0}^{1} \boldsymbol{\nu}^{\mathsf{a}-1} Y^{\ast} (\boldsymbol{\nu}s + (1-\boldsymbol{\nu})t) \times \mathfrak{G}^{\ast} (\boldsymbol{\nu}s + (1-\boldsymbol{\nu})t) \\ &+ \boldsymbol{\nu}^{\mathsf{a}-1} Y^{\ast} ((1-\boldsymbol{\nu})s + \boldsymbol{\nu}t) \times \mathfrak{G}^{\ast} ((1-\boldsymbol{\nu})s + \boldsymbol{\nu}t) d\boldsymbol{\nu} \\ &\leq q^{\ast}(s,t) \int\_{0}^{1} \boldsymbol{\nu}^{\mathsf{a}-1} \Big[\boldsymbol{\nu}^{2} + (1-\boldsymbol{\nu})^{2}\Big] d\boldsymbol{\nu} + 2\nabla^{\ast}(s,t) \int\_{0}^{1} \boldsymbol{\nu}^{\mathsf{a}-1} \boldsymbol{\nu} (1-\boldsymbol{\nu}) d\boldsymbol{\nu}. \end{split}$$

It follows that

Γ(a) (*t*−*s*) a - I a *s*+ *Y*∗(*t*) × G∗(*t*) + I a *t*− *Y*∗(*s*) × G∗(*s*) ≤ 2 a 1 2 − a (a+1)(a+2) *ϕ*∗(*s*, *t*) + 2 a a (a+1)(a+2) ∇∗(*s*, *t*) Γ(a) (*t*−*s*) a - I a *s*+ *Y* ∗ (*t*) × G∗ (*t*) + I a *t*− *Y* ∗ (*s*) × G∗ (*s*) ≤ 2 a 1 2 − a (a+1)(a+2) *ϕ* ∗ (*s*, *t*) + 2 a a (a+1)(a+2) ∇∗ (*s*, *t*), Γ(a) (*t*−*s*) a - I a *s*+ *Y*∗(*t*) × G∗(*t*) + I a *t*− *Y*∗(*s*) × G∗(*s*) ≤ 2 a 1 2 − a (a+1)(a+2) *ϕ*∗(*s*, *t*) + 2 a a (a+1)(a+2) ∇∗(*s*, *t*) Γ(a) (*t*−*s*) a - I a *s*+ *Y* ∗ (*t*) × G∗ (*t*) + I a *t*− *Y* ∗ (*s*) × G∗ (*s*) ≤ 2 a 1 2 − a (a+1)(a+2) *ϕ* ∗ (*s*, *t*) + 2 a a (a+1)(a+2) ∇∗ (*s*, *t*),

That is,

$$\begin{split} & \frac{\Gamma(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} \Big[ \mathcal{Z}\_{s^{+}}^{\mathfrak{a}} \, \mathcal{Y}\_{\*}(t) \times \mathfrak{G}\_{\*}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \, \mathcal{Y}\_{\*}(s) \times \mathfrak{G}\_{\*}(s) \, \mathcal{Z}\_{s^{+}}^{\mathfrak{a}} \, \mathcal{Y}^{\*}(t) \times \mathfrak{G}^{\*}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \, \mathcal{Y}^{\*}(s) \times \mathfrak{G}^{\*}(s) \Big] \\ & \leq p \, \frac{2}{\mathfrak{a}} \Big( \frac{1}{2} - \frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)} \Big[ \!\vert \boldsymbol{\varrho}\_{\*}(\mathsf{s},t) \ \ \boldsymbol{\varrho}^{\*}(\mathsf{s},t) \big] + \frac{2}{\mathfrak{a}} \Big( \frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)} \Big[ \!\vert \nabla\_{\mathrm{s}}(\mathsf{s},t) \ \ \nabla^{\mathfrak{s}}(\mathsf{s},t) \right] \end{split}$$

Thus,

$$\begin{aligned} \frac{\overline{I(\mathfrak{a})}}{2(t-s)^{\mathfrak{a}}} \left[ \mathcal{Z}\_{s^{+}}^{\mathfrak{a}} \, \, Y(t) \times \mathfrak{G}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \, Y(s) \times \mathfrak{G}(s) \right] \\ \leq\_{p} & \left( \frac{1}{2} - \frac{\mathfrak{a}}{(a+1)(a+2)} \right) \varphi(s,t) + \left( \frac{\mathfrak{a}}{(a+1)(a+2)} \right) \nabla(s,t) . \end{aligned}$$

and the theorem has been established.

**Example 2.** *Let* [*s*, *t*] = [0, 2 ]*,* a = 1 2 , *Y*(*ω*) = *ω* 2 , 3*ω* 2 *, and* G(*ω*) = [*ω*, 3*ω*]. *Since the left and right endpoint functions, Y*∗(*ω*) = *ω* 2 , *Y* ∗ (*ω*) = 3*ω* 2 *,* G∗(*ω*) = *ω and* G∗ (*ω*) = 3*ω, are left and right convex functions, then Y*(*ω*) *and* G(*ω*) *are both left and right convex I*·*V-Fs. We clearly see that Y*(*ω*) × G(*ω*) ∈ *L* [*s*, *t*], X + *I , and*

$$\begin{split} &\frac{\Gamma(1+\mathfrak{a})}{2(t-s)^{\mathfrak{a}}} \left[ \mathcal{T}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \, \mathcal{Y}\_{\ast}(t) \times \mathfrak{G}\_{\ast}(t) + \mathcal{T}\_{\mathfrak{t}^{-}}^{\mathfrak{a}} \, \mathcal{Y}\_{\ast}(s) \times \mathfrak{G}\_{\ast}(s) \right] \\ = & \frac{\Gamma(\frac{3}{2})}{2\sqrt{2}} \frac{1}{\sqrt{\pi}} \int\_{0}^{2} (2-\omega)^{\frac{-1}{2}} \left( \frac{1}{2}\omega^{2} \right) d\omega + \frac{\Gamma(\frac{3}{2})}{2\sqrt{2}} \frac{1}{\sqrt{\pi}} \int\_{0}^{2} (\omega)^{\frac{-1}{2}} \left( \frac{1}{2}\omega^{2} \right) d\omega \approx 0.7333 \,\mu\text{V} \\ & \frac{\Gamma(1+\mathfrak{a})}{2(t-s)^{\mathfrak{a}}} \left[ \mathcal{T}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \, \mathcal{Y}^{\ast}(t) \times \mathfrak{G}^{\ast}(t) + \mathcal{T}\_{\mathfrak{t}^{-}}^{\mathfrak{a}} \, \mathcal{Y}^{\ast}(s) \times \mathfrak{G}^{\ast}(s) \right] \\ = & \frac{\Gamma(\frac{3}{2})}{2\sqrt{2}} \frac{1}{\sqrt{\pi}} \int\_{0}^{2} (2-\omega)^{\frac{-1}{2}} \, \frac{9}{2} \omega^{2} d\omega + \frac{\Gamma(\frac{3}{2})}{2\sqrt{2}} \frac{1}{\sqrt{\pi}} \int\_{0}^{2} (\omega)^{\frac{-1}{2}} \, \frac{9}{2} \omega^{2} d\omega \approx 6.5997, \end{split}$$

*Note that*

$$\begin{cases} \frac{1}{2} - \frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)} \Big[ \mathfrak{q}\_{\*}(\mathfrak{s},t) = \left[ \mathcal{Y}\_{\*}(\mathfrak{s}) \times \mathfrak{G}\_{\*}(\mathfrak{s}) + \mathcal{Y}\_{\*}(t) \times \mathfrak{G}\_{\*}(t) \right] = \frac{11}{15}, \\ \frac{1}{2} - \frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)} \Big] \mathcal{q}^{\*}(\mathfrak{s},t) = \left[ \mathcal{Y}^{\*}(\mathfrak{s}) \times \mathfrak{G}^{\*}(\mathfrak{s}) + \mathcal{Y}^{\*}(t) \times \mathfrak{G}^{\*}(t) \right] = \frac{33}{5}, \\ \left( \frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)} \right) \nabla\_{\ast}(\mathfrak{s},t) = \left[ \mathcal{Y}\_{\*}(\mathfrak{s}) \times \mathfrak{G}\_{\*}(t) + \mathcal{Y}\_{\*}(t) \times \mathfrak{G}\_{\*}(\mathfrak{s}) \right] = \frac{2}{15}(0), \\ \left( \frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)} \right) \nabla\_{\ast}(\mathfrak{s},t) = \left[ \mathcal{Y}^{\*}(\mathfrak{s}) \times \mathfrak{G}^{\*}(t) + \mathcal{Y}^{\*}(t) \times \mathfrak{G}^{\*}(\mathfrak{s}) \right] = \frac{2}{15}(0). \end{cases}$$

*Therefore, we have*

$$\begin{array}{c} \left(\frac{1}{2} - \frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)}\right) \varphi(\mathfrak{s},t) + \left(\frac{\mathfrak{a}}{(\mathfrak{a}+1)(\mathfrak{a}+2)}\right) \nabla(\mathfrak{s},t) \\ = \begin{bmatrix} \frac{11}{15}\frac{33}{5} \end{bmatrix} + \frac{2}{15}[0,0] = \begin{bmatrix} \frac{11}{15}\frac{33}{5} \end{bmatrix} \end{array}$$

*It follows that*

$$[0.7333, \ 6.5997] \leq\_p \left[\frac{11}{15}, \frac{33}{5}\right]$$

*and Theorem 4 has been demonstrated*.

**Theorem 5.** *Let Y*, G : [*s*, *t*] → X + *I be two left and right convex I*·*V-Fs, provided by Y*(*ω*) = [*Y*∗(*ω*), *Y* ∗ (*ω*)] *and* G(*ω*) = [G∗(*ω*), G∗ (*ω*)] *for all ω* ∈ [*s*, *t*]*. If Y* × G ∈ *L* [*s*, *t*], X + *I , then*

$$\begin{split} \frac{1}{a} \, ^\mathsf{T} \Big( \frac{s+t}{2} \big) \times \mathfrak{G} \Big( \frac{s+t}{2} \big) \leq\_p \frac{\mathsf{I} (\mathsf{a}+1)}{4(t-s)^\mathsf{a}} \Big[ \mathscr{Z}\_{s^+}^\mathsf{a} \, ^\mathsf{Y} (t) \times \mathfrak{G} (t) + \mathscr{Z}\_{t^-}^\mathsf{a} \, ^\mathsf{Y} (s) \times \mathfrak{G} (s) \Big] \\ + \frac{1}{2a} \Big( \frac{1}{2} - \frac{\mathsf{a}}{(\mathsf{a}+1)(\mathsf{a}+2)} \Big) \nabla (s,t) + \frac{1}{2a} \Big( \frac{\mathsf{a}}{(\mathsf{a}+1)(\mathsf{a}+2)} \Big) \varrho (s,t) \, \end{split}$$

*where ϕ*(*s*, *t*) = *Y*(*s*) × G(*s*) + *Y*(*t*) × G(*t*), ∇(*s*, *t*) = *Y*(*s*) × G(*t*) + *Y*(*t*) × G(*s*), *ϕ*(*s*, *t*) = [*ϕ*∗(*s*, *t*), *ϕ* ∗ (*s*, *t*)], *and* ∇(*s*, *t*) = [∇∗(*s*, *t*), ∇∗ (*s*, *t*)].

**Proof.** Consider that *Y*, G : [*s*, *t*] → X + *I* are left and right convex *I*·*V-Fs*. Then, by hypothesis, we have *s*+*t s*+*t*

*Y*∗ 2 × G∗ 2 *Y* ∗ *s*+*t* 2 × G∗ *s*+*t* 2 ≤ 1 4 *Y*∗(*νs* + (1 − *ν*)*t*) × G∗(*νs* + (1 − *ν*)*t*) +*Y*∗(*νs* + (1 − *ν*)*t*) × G∗((1 − *ν*)*s* + *νt*) +1 4 *Y*∗((1 − *ν*)*s* + *νt*) × G∗(*νs* + (1 − *ν*)*t*) +*Y*∗((1 − *ν*)*s* + *νt*) × G∗((1 − *ν*)*s* + *νt*) ≤ 1 4 *Y* ∗ (*νs* + (1 − *ν*)*t*) × G∗ (*νs* + (1 − *ν*)*t*) +*Y* ∗ (*νs* + (1 − *ν*)*t*) × G∗ ((1 − *ν*)*s* + *νt*) +1 4 *Y* ∗ ((1 − *ν*)*s* + *νt*) × G∗ (*νs* + (1 − *ν*)*t*) +*Y* ∗ ((1 − *ν*)*s* + *νt*) × G∗ ((1 − *ν*)*s* + *νt*) , ≤ 1 4 *Y*∗(*νs* + (1 − *ν*)*t*) × G∗(*νs* + (1 − *ν*)*t*) +*Y*∗((1 − *ν*)*s* + *νt*) × G∗((1 − *ν*)*s* + *νt*) +1 4 (*νY*∗(*s*) + (1 − *ν*)*Y*∗(*t*)) ×((1 − *ν*)G∗(*s*) + *ν*G∗(*t*)) +((1 − *ν*)*Y*∗(*s*) + *νY*∗(*t*)) ×(*ν*G∗(*s*) + (1 − *ν*)G∗(*t*)) ≤ 1 4 *Y* ∗ (*νs* + (1 − *ν*)*t*) × G∗ (*νs* + (1 − *ν*)*t*) +*Y* ∗ ((1 − *ν*)*s* + *νt*) × G∗ ((1 − *ν*)*s* + *νt*) +1 4 (*νY* ∗ (*s*) + (1 − *ν*)*Y* ∗ (*t*)) ×((1 − *ν*)G∗ (*s*) + *ν*G∗ (*t*)) +((1 − *ν*)*Y* ∗ (*s*) + *νY* ∗ (*t*)) ×(*ν*G∗ (*s*) + (1 − *ν*)G∗ (*t*)) , = 1 4 *Y*∗(*νs* + (1 − *ν*)*t*) × G∗(*νs* + (1 − *ν*)*t*) +*Y*∗((1 − *ν*)*s* + *νt*) × G∗((1 − *ν*)*s* + *νt*) +1 4 " n *ν* 2 + (1 − *ν*) 2 o ∇∗(*s*, *t*) +{*ν*(1 − *ν*) + (1 − *ν*)*ν*}*ϕ*∗(*s*, *t*) # = 1 4 *Y* ∗ (*νs* + (1 − *ν*)*t*) × G∗ (*νs* + (1 − *ν*)*t*) +*Y* ∗ ((1 − *ν*)*s* + *νt*) × G∗ ((1 − *ν*)*s* + *νt*) +1 4 " n *ν* 2 + (1 − *ν*) 2 o ∇∗ (*s*, *t*) +{*ν*(1 − *ν*) + (1 − *ν*)*ν*}*ϕ* ∗ (*s*, *t*) # .

The upcoming results discuss the H–H Fejér type inequality left and right convex *I*·*V-F.* Firstly, we achieve second H–H Fejér type inequality.

**Theorem 6.** *Let Y* : [*s*, *t*] → X + *I be a left and right convex I*·*V-F, with s* < *t, provided by Y*(*ω*) = [*Y*∗(*ω*), *Y* ∗ (*ω*)] *for all ω* ∈ [*s*, *t*]*. Let Y* ∈ *L* [*s*, *t*], X + *I and* C : [*s*, *t*] → R, C(*ω*) ≥ 0, *be symmetric with respect to s*+*t* 2 .*Then,*

$$\mathbb{E}\left[\mathcal{Z}\_{\mathbf{s}^+}^{\mathbf{a}}\,\,\mathbf{Y}\mathfrak{C}(t) + \mathcal{Z}\_{\mathbf{t}^-}^{\mathbf{a}}\,\,\mathbf{Y}\mathfrak{C}(s)\right] \leq\_p \frac{\mathsf{Y}(s) + \mathsf{Y}(t)}{2} \left[\mathcal{Z}\_{\mathbf{s}^+}^{\mathbf{a}}\,\,\mathfrak{C}(t) + \mathcal{Z}\_{\mathbf{t}^-}^{\mathbf{a}}\,\,\mathfrak{C}(s)\right] \tag{13}$$

*If Y is a concave I*·*V-F, then inequality (13) is reversed.*

**Proof.** Let *Y* be a left and right convex *I*·*V-F* and *ν* a−1C(*νs* + (1 − *ν*)*t*) ≥ 0. Then, we have

$$\begin{array}{l} \nu^{a-1}Y\_\*(\upsilon s + (1-\upsilon)t)\mathbb{C}(\upsilon s + (1-\upsilon)t) \\ \qquad \le \nu^{a-1}(\upsilon Y\_\*(s) + (1-\upsilon)Y\_\*(t))\mathbb{C}(\upsilon s + (1-\upsilon)t) \\ \nu^{a-1}Y^\*(\upsilon s + (1-\upsilon)t)\mathbb{C}(\upsilon s + (1-\upsilon)t) \\ \qquad \le \nu^{a-1}(\upsilon Y^\*(s) + (1-\upsilon)Y^\*(t))\mathbb{C}(\upsilon s + (1-\upsilon)t). \end{array} \tag{14}$$

and

*ν* a−1*Y*∗((1 − *ν*)*s* + *νt*)*C*((1 − *ν*)*s* + *νt*) ≤ *ν* a−1 ((1 − *ν*)*Y*∗(*s*) + *νY*∗(*t*))*C*((1 − *ν*)*s* + *νt*) *ν* a−1*Y* ∗ ((1 − *ν*)*s* + *νt*)C((1 − *ν*)*s* + *νt*) ≤ *ν* a−1 ((1 − *ν*)*Y* ∗ (*s*) + *νY* ∗ (*t*))C((1 − *ν*)*s* + *νt*).

Since C is symmetric, then

$$\begin{array}{l} = \left[Y\_{\ast}(s) + Y\_{\ast}(t)\right] \int\_{0}^{1} \nu^{\mathfrak{a}-1} \mathbb{C}((1-\nu)s + \nu t) \, d\nu\\ = \left[Y^{\ast}(s) + Y^{\ast}(t)\right] \int\_{0}^{1} \nu^{\mathfrak{a}-1} \mathbb{C}((1-\nu)s + \nu t) \, d\nu. \end{array}$$

$$\begin{split} \mathcal{I} &= \frac{Y\_{\ast}(s) + Y\_{\ast}(t)}{2} \frac{\overline{I(\mathfrak{a})}}{(t-s)^{\mathfrak{a}}} \Big[ \mathcal{Z}\_{\mathbb{s}^{+}}^{\mathfrak{a}} \, \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \, \mathfrak{C}(s) \Big], \\ &= \frac{Y^{\ast}(s) + Y^{\ast}(t)}{2} \frac{\overline{I(\mathfrak{a})}}{(t-s)^{\mathfrak{a}}} \Big[ \mathcal{Z}\_{\mathbb{s}^{+}}^{\mathfrak{a}} \, \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \, \mathfrak{C}(s) \Big]. \end{split} \tag{16}$$

Since

$$\begin{split} \int\_{0}^{1} \nu^{a-1} Y\_{\*} (\nu s + (1-\nu)t) \mathfrak{C}((1-\nu)s + \nu t) d\upsilon \\ &\quad + \int\_{0}^{1} \nu^{a-1} Y\_{\*} ((1-\nu)s + \nu t) \mathfrak{C}((1-\nu)s + \nu t) d\upsilon \\ &= \frac{1}{(t-s)^{a}} \int\_{s}^{t} (\omega - s)^{a-1} Y\_{\*} (s + t - \omega) \mathfrak{C}(\omega) d\omega \\ &= \frac{1}{(t-s)^{a}} \int\_{s}^{t} (\omega - s)^{a-1} Y\_{\*} (\omega) \mathfrak{C}(\omega) d\omega \\ &\quad + \frac{1}{(t-s)^{a}} \int\_{s}^{t} (\omega - s)^{a-1} Y\_{\*} (\omega) \mathfrak{C}(\omega) d\omega \\ &= \frac{\Gamma(a)}{(t-s)^{a}} \left[ T\_{\mathfrak{C}}^{a} + Y\_{\*} \mathfrak{C}(t) + \mathcal{T}\_{t}^{a} \cdot Y\_{\*} \mathfrak{C}(s) \right], \\ \int\_{0}^{1} \nu^{a-1} Y^{\*} (\nu s + (1-\nu)t) \mathfrak{C}((1-\nu)s + \nu t) d\nu \\ &\quad + \int\_{0}^{1} \nu^{a-1} Y^{\*} ((1-\nu)s + \nu t) \mathfrak{C}((1-\nu)s + \nu t) d\nu \\ &= \frac{\Gamma(a)}{(t-s)^{a}} \left[ T\_{\mathfrak{C}}^{a} + Y^{\*} \mathfrak{C}(t) + \mathcal{T}\_{t-}^{a} \cdot Y^{\*} \mathfrak{C}(s) \right]. \end{split} \tag{17}$$

then, from (16), we have

Γ(a) (*t*−*s*) a - I a *s*+ *Y* ∗C(*t*) + I a *t*− *Y* ∗C(*s*) ≤ *Y* ∗ (*s*)+*Y* ∗ (*t*) 2 Γ(a) (*t*−*s*) a - I a *s*+ C(*t*) + I a *t*− C(*s*) ≤ *Y* ∗ (*s*)+*Y* ∗ (*t*) 2 Γ(a) (*t*−*s*) a - I a *s*+ C(*t*) + I a *t*− C(*s*) , Γ(a) (*t*−*s*) a - I a *s*+ *Y* ∗C(*t*) + I a *t*− *Y* ∗C(*s*) ≤ *Y* ∗ (*s*)+*Y* ∗ (*t*) 2 Γ(a) (*t*−*s*) a - I a *s*+ C(*t*) + I a *t*− C(*s*) ≤ *Y* ∗ (*s*)+*Y* ∗ (*t*) 2 Γ(a) (*t*−*s*) a - I a *s*+ C(*t*) + I a *t*− C(*s*) ,

That is,

$$\frac{\Gamma(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} \left[ \left[ \mathcal{Z}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \, \, \mathcal{Y}\_{\ast} \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \, \, \mathcal{Y}\_{\ast} \mathfrak{C}(s) \right], \, \mathcal{Z}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \, \, \mathcal{Y}^{\ast} \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \, \, \mathcal{Y}^{\ast} \mathfrak{C}(s) \right].$$

$$\leq\_{p} \frac{\Gamma(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} \left[ \frac{Y\_{\ast}(\mathfrak{s}) + Y\_{\ast}(t)}{2}, \frac{Y^{\ast}(\mathfrak{s}) + Y^{\ast}(t)}{2} \right] \left[ \mathcal{Z}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \mathfrak{C}(\mathfrak{s}) \right],$$

$$\leq\_{p} \frac{\Gamma(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} \left[ \frac{Y\_{\ast}(\mathfrak{s}) + Y\_{\ast}(t)}{2}, \frac{Y^{\ast}(\mathfrak{s}) + Y^{\ast}(t)}{2} \right] \left[ \mathcal{Z}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \mathfrak{C}(\mathfrak{s}) \right],$$

Hence,

$$\left[\mathcal{Z}\_{s^+}^{\mathbf{a}} \, \mathcal{Y}\mathfrak{C}(t) \leq\_p \mathcal{Z}\_{t^-}^{\mathbf{a}} \, \mathcal{Y}\mathfrak{C}(s)\right] \leq\_p \frac{\mathcal{Y}(s) + \mathcal{Y}(t)}{2} \left[\mathcal{Z}\_{s^+}^{\mathbf{a}} \, \mathfrak{C}(t) + \mathcal{Z}\_{t^-}^{\mathbf{a}} \, \mathfrak{C}(s)\right].$$

$$\leq\_p \frac{\mathcal{Y}(s) + \mathcal{Y}(t)}{2} \left[\mathcal{Z}\_{s^+}^{\mathbf{a}} \, \mathfrak{C}(t) + \mathcal{Z}\_{t^-}^{\mathbf{a}} \, \mathfrak{C}(s)\right].$$

Now, we first obtain the H–H Fejér type inequality for the left and right convex *I*·*V-F*.

**Theorem 7.** *Let Y* : [*s*, *t*] → X + *I be a left and right convex I*·*V-F, with s* < *t, and defined by Y*(*ω*) = [*Y*∗(*ω*), *Y* ∗ (*ω*)] *for all ω* ∈ [*s*, *t*]*. If Y* ∈ *L* [*s*, *t*], X + *I and* C : [*s*, *t*] → R, C(*ω*) ≥ 0 *are symmetric with respect to s*+*t* 2 *, then*

$$\mathcal{Y}\left(\frac{s+t}{2}\right)\left[\mathcal{Z}\_{\mathbf{s}^+}^{\mathbf{a}}\,\mathfrak{C}(t)+\mathcal{Z}\_{t^-}^{\mathbf{a}}\,\mathfrak{C}(s)\right] \leq\_p \left[\mathcal{Z}\_{\mathbf{s}^+}^{\mathbf{a}}\,\mathcal{Y}\mathfrak{C}(t)+\mathcal{Z}\_{t^-}^{\mathbf{a}}\,\mathcal{Y}\mathfrak{C}(s)\right] \tag{18}$$

*If Y is a concave I*·*V-F, then inequality (18) is reversed*.

**Proof.** Since *Y* is a left and right convex *I*·*V-F*, then we have

$$\begin{array}{l} \chi\_{\*}\left(\frac{s+t}{2}\right) \leq \frac{1}{2} (\chi\_{\*}(\upsilon s + (1-\nu)t) + \chi\_{\*}((1-\nu)s + \upsilon t)) \\ \chi^{\*}\left(\frac{s+t}{2}\right) \leq \frac{1}{2} (\chi^{\*}(\upsilon s + (1-\nu)t) + \chi^{\*}((1-\nu)s + \upsilon t)), \end{array} \tag{19}$$

Since C(*νs* + (1 − *ν*)*t*) = C((1 − *ν*)*s* + *νt*), then by multiplying (19) by *ν* a−1C ((1 − *ν*)*s* + *νt*) and integrating it, with respect to *ν* over [0, 1], we obtain

$$\begin{split} \mathcal{Y}\_{\boldsymbol{s}}\left(\frac{s+t}{2}\right) \int\_{0}^{1} \boldsymbol{\nu}^{a-1} \mathfrak{C}((1-\boldsymbol{\nu})s+\boldsymbol{\nu}t) d\boldsymbol{\nu} \\ \leq & \frac{1}{2} \Big( \int\_{0}^{1} \boldsymbol{\nu}^{a-1} Y\_{\ast}(\boldsymbol{\nu}s+(1-\boldsymbol{\nu})t) \mathfrak{C}((1-\boldsymbol{\nu})s+\boldsymbol{\nu}t) d\boldsymbol{\nu} \\ \quad + \int\_{0}^{1} \boldsymbol{\nu}^{a-1} Y\_{\ast}((1-\boldsymbol{\nu})s+\boldsymbol{\nu}t) \mathfrak{C}((1-\boldsymbol{\nu})s+\boldsymbol{\nu}t) d\boldsymbol{\nu} \Big) \\ \leq & \frac{1}{2} \Big( \int\_{0}^{1} \boldsymbol{\nu}^{a-1} Y^{\ast}(\boldsymbol{\nu}s+(1-\boldsymbol{\nu})t) \mathfrak{C}((1-\boldsymbol{\nu})s+\boldsymbol{\nu}t) d\boldsymbol{\nu} \\ \leq & \frac{1}{2} \Big( \quad + \int\_{0}^{1} \boldsymbol{\nu}^{a-1} Y^{\ast}((1-\boldsymbol{\nu})s+\boldsymbol{\nu}t) \mathfrak{C}((1-\boldsymbol{\nu})s+\boldsymbol{\nu}t) d\boldsymbol{\nu} \Big) \end{split} \tag{20}$$

Let *ω* = (1 − *ν*)*s* + *νt*. Then, we have

$$\begin{split} \int\_{0}^{1} v^{a-1} Y\_{\*} (\nu s + (1-\nu)t) \mathfrak{C}((1-\nu)s + \nu t) d\upsilon \\ &\quad + \int\_{0}^{1} v^{a-1} Y\_{\*} ((1-\nu)s + \nu t) \mathfrak{C}((1-\nu)s + \nu t) d\upsilon \\ &= \frac{1}{(t-s)^{a}} \int\_{s}^{t} (\omega - s)^{a-1} Y\_{\*} (s + t - \omega) \mathfrak{C}(\omega) d\omega \\ &\quad + \frac{1}{(t-s)^{a}} \int\_{s}^{t} (\omega - s)^{a-1} Y\_{\*} (\omega) \mathfrak{C}(\omega) d\omega \\ &= \frac{1}{(t-s)^{a}} \int\_{s}^{t} (\omega - s)^{a-1} Y\_{\*} (\omega) \mathfrak{C}(s + t - \omega) d\omega \\ &\quad + \frac{1}{(t-s)^{a}} \int\_{s}^{t} (\omega - s)^{a-1} Y\_{\*} (\omega) \mathfrak{C}(\omega) d\omega \\ &= \frac{\Gamma(a)}{(t-s)^{a}} \left[ Z\_{\*}^{a}, Y\_{\*} \mathfrak{C}(t) + Z\_{t}^{a}, Y\_{\*} \mathfrak{C}(s) \right], \\ \int\_{0}^{1} v^{a-1} Y^{\*} (\nu s + (1-\nu)t) \mathfrak{C}((1-\nu)s + \nu t) d\nu \\ &\quad + \int\_{0}^{1} \nu^{a-1} Y^{\*} ((1-\nu)s + \nu t) \mathfrak{C}((1-\nu)s + \nu t) d\nu \\ &= \frac{\Gamma(a)}{(t-s)^{a}} \left[ Z\_{\*}^{a}, Y^{\*} \mathfrak{C}(t) + Z\_{t}^{a} \cdot Y^{\*} \mathfrak{C}(s) \right]. \end{split} \tag{21}$$

Then, from (21), we have

$$\begin{split} \frac{\overline{\varGamma}(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} Y\_{\*} \left( \frac{s+t}{2} \right) \left[ \mathscr{Z}\_{\mathrm{s}^{+}}^{\mathfrak{a}} \mathfrak{C}(t) + \mathscr{Z}\_{t^{-}}^{\mathfrak{a}} \mathfrak{C}(s) \right] \\ \leq & \frac{\varGamma(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} \left[ \mathscr{Z}\_{\mathrm{s}^{+}}^{\mathfrak{a}} \operatorname{ Y}\_{\*} \mathfrak{C}(t) + \mathscr{Z}\_{t^{-}}^{\mathfrak{a}} \operatorname{ Y}\_{\*} \mathfrak{C}(s) \right] \\ \frac{\varGamma(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} Y^{\*} \left( \frac{s+t}{2} \right) \left[ \mathscr{Z}\_{\mathrm{s}^{+}}^{\mathfrak{a}} \mathfrak{C}(t) + \mathscr{Z}\_{t^{-}}^{\mathfrak{a}} \mathfrak{C}(s) \right] \\ \leq & \frac{\varGamma(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} \left[ \mathscr{Z}\_{\mathrm{s}^{+}}^{\mathfrak{a}} \operatorname{ Y}^{\mathfrak{a}} \mathfrak{C}(t) + \mathscr{Z}\_{t^{-}}^{\mathfrak{a}} \operatorname{ Y}^{\mathfrak{a}} \mathfrak{C}(s) \right] \end{split}$$

from which, we have

$$\begin{array}{ll} \frac{\overline{I}(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} \left[ \mathcal{Y}\_{\*} \left( \frac{s+t}{2} \right), \mathcal{Y}^{\*} \left( \frac{s+t}{2} \right) \right] \left[ \mathcal{Z}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \mathfrak{C}(t) + \mathcal{Z}\_{\mathfrak{t}^{-}}^{\mathfrak{a}} \mathfrak{C}(s) \right] \\ \leq & \frac{\overline{I}(\mathfrak{a})}{p \left(t-s\right)^{\mathfrak{a}}} \left[ \mathcal{Z}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \mathcal{Y}\_{\*} \mathfrak{C}(t) + \mathcal{Z}\_{\mathfrak{t}^{-}}^{\mathfrak{a}} \mathcal{Y}\_{\*} \mathfrak{C}(s), \mathcal{Z}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \mathcal{Y}^{\*} \mathfrak{C}(t) + \mathcal{Z}\_{\mathfrak{t}^{-}}^{\mathfrak{a}} \mathcal{Y}^{\*} \mathfrak{C}(s) \right], \end{array}$$

That is,

$$\frac{\Gamma(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} Y(\frac{s+t}{2}) \left[ \mathcal{Z}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \mathfrak{C}(s) \right],$$

$$\leq\_{p} \frac{\Gamma(\mathfrak{a})}{(t-s)^{\mathfrak{a}}} \left[ \mathcal{Z}\_{\mathfrak{s}^{+}}^{\mathfrak{a}} \ Y \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathfrak{a}} \ Y \mathfrak{C}(s) \right].$$

This completes the proof.

**Example 3.** *We consider the I*·*V-F Y* : [0, 2] → X + *I , defined by Y*(*ω*) = -2 − √ *ω* , 2 2 − √ *ω . Since endpoint functions Y*∗(*ω*), *Y* ∗ (*ω*) *are convex functions, then Y*(*ω*) *is a left and right convex I*·*V-F. If*

$$\mathfrak{C}(\omega) = \left\{ \begin{array}{c} \sqrt{\omega} \\ \sqrt{2 - \omega} \end{array} \begin{array}{c} \omega \in [0, 1] \\ \omega \in (1, 2] \end{array} \right\}$$

*then* C(2 − *ω*) = C(*ω*) ≥ 0 *for all ω* ∈ [0, 2]. *Since Y*∗(*ω*) = 2 − √ *ω and Y* ∗ (*ω*) = 2 2 − √ *ω* , *if* a = 1 2 , *then we compute the following:*

$$\begin{aligned} \left[\mathbb{Z}\_{s^+}^{\mathbf{a}} \, \mathbf{Y} \mathfrak{C}(t) \tilde{+} \mathbb{Z}\_{t^-}^{\mathbf{a}} \, \mathbf{Y} \mathfrak{C}(s)\right] &\leq\_p \frac{Y(s) + Y(t)}{2} \left[\mathbb{Z}\_{s^+}^{\mathbf{a}} \, \mathfrak{C}(t) + \mathbb{Z}\_{t^-}^{\mathbf{a}} \, \mathfrak{C}(s)\right] \\\\ \frac{Y\_\*(s) + Y\_\*(t)}{2} \left[\mathbb{Z}\_{s^+}^{\mathbf{a}} \, \mathfrak{C}(t) + \mathbb{Z}\_{t^-}^{\mathbf{a}} \, \mathfrak{C}(s)\right] &= \frac{\pi}{\sqrt{2}} \binom{4 - \sqrt{2}}{2} \\\\ \frac{Y^\*(s) + Y^\*(t)}{2} \left[\mathbb{Z}\_{s^+}^{\mathbf{a}} \, \mathfrak{C}(t) + \mathbb{Z}\_{t^-}^{\mathbf{a}} \, \mathfrak{C}(s)\right] &= \frac{\pi}{\sqrt{2}} \binom{4 - \sqrt{2}}{2}, \end{aligned} \tag{22}$$

$$\begin{array}{c} \frac{Y\_\*(s) + Y\_\*(t)}{2} \left[ \mathcal{Z}\_{s^+}^{\mathfrak{a}} \mathfrak{C}(t) + \mathcal{Z}\_{t^-}^{\mathfrak{a}} \mathfrak{C}(s) \right] = \frac{\pi}{\sqrt{2}} \binom{4 - \sqrt{2}}{2} \\\ \frac{Y^\*(s) + Y^\*(t)}{2} \left[ \mathcal{Z}\_{s^+}^{\mathfrak{a}} \mathfrak{C}(t) + \mathcal{Z}\_{t^-}^{\mathfrak{a}} \mathfrak{C}(s) \right] = \frac{\pi}{\sqrt{2}} \binom{4 - \sqrt{2}}{2} . \end{array} \tag{23}$$

$$\begin{aligned} \left[ \mathcal{Z}\_{s^+}^{\mathfrak{a}} \, \, \mathcal{Y}\_\* \mathfrak{C}(t) + \mathcal{Z}\_{t^-}^{\mathfrak{a}} \, \, \mathcal{Y}\_\* \mathfrak{C}(s) \right] &= \frac{1}{\sqrt{\pi}} \Big( 2\pi + \frac{4 - 8\sqrt{2}}{3} \Big), \\\left[ \mathcal{Z}\_{s^+}^{\mathfrak{a}} \, \, \mathcal{Y}^\* \mathfrak{C}(t) + \mathcal{Z}\_{t^-}^{\mathfrak{a}} \, \, \mathcal{Y}^\* \mathfrak{C}(s) \right] &= \frac{2}{\sqrt{\pi}} \Big( 2\pi + \frac{4 - 8\sqrt{2}}{3} \Big). \end{aligned} \tag{24}$$

*From (22)–(24), (13) we have*

$$\frac{1}{\sqrt{\pi}} \left[ \left( 2\pi + \frac{4 - 8\sqrt{2}}{3} \right) , 2 \left( 2\pi + \frac{4 - 8\sqrt{2}}{3} \right) \right] \\ \quad \leq \, \,\_p \frac{\pi}{\sqrt{2}} \left[ \frac{4 - \sqrt{2}}{2} , 4 - \sqrt{2} \right] = \frac{\pi}{\sqrt{2}} \left[ \frac{4 - \sqrt{2}}{2} , 4 - \sqrt{2} \right] .$$

*Hence, Theorem 6 is verified. For Theorem 7, we have*

$$
\begin{bmatrix}
\Upsilon\_\* \left( \frac{s+t}{2} \right) \left[ \mathcal{Z}\_{s^+}^{\mathbf{a}} \mathfrak{C}(t) + \mathcal{Z}\_{t^-}^{\mathbf{a}} \mathfrak{C}(s) \right] = \sqrt{\pi}, \\
\mathcal{Y}^\* \left( \frac{s+t}{2} \right) \left[ \mathcal{Z}\_{s^+}^{\mathbf{a}} \mathfrak{C}(t) + \mathcal{Z}\_{t^-}^{\mathbf{a}} \mathfrak{C}(s) \right] = \mathfrak{D} \sqrt{\pi}.
\end{bmatrix} \tag{25}
$$

*From (24) and (25), we have*

$$2\sqrt{\pi}[1,2] \quad \le \ \_ {p} \frac{1}{\sqrt{\pi}} \left[2\pi + \frac{4-8\sqrt{2}}{3}, \ 2\left(2\pi + \frac{4-8\sqrt{2}}{3}\right)\right]$$

*Hence, (18) has been verified.*

**Remark 3.** *If one takes* C(*ω*) = 1*, then, from (13) and (18), we acquire (5). Let us take* a = 1. *Then, we achieve the coming inequality (see [22])*.

$$\mathbb{Y}\left(\frac{s+t}{2}\right) \leq\_p \frac{1}{\int\_s^t \mathfrak{C}(\omega)d\omega} \int\_s^t \mathbb{Y}(\omega)\mathfrak{C}(\omega)d\omega \leq\_p \frac{\mathbb{Y}(s)+\mathbb{Y}(t)}{2}.$$

*If we take Y*∗(*ω*) = *Y* ∗ (*ω*), *then from (13) and (18), we acquire the coming inequality (see [33])*.

$$Y\left(\frac{s+t}{2}\right)\left[\mathcal{Z}\_{s^{+}}^{\mathbf{a}}\ \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathbf{a}}\ \mathfrak{C}(s)\right] \leq\_{p} \left[\mathcal{Z}\_{s^{+}}^{\mathbf{a}}\ \ Y\mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathbf{a}}\ \ Y\mathfrak{C}(s)\right],$$

$$\leq\_{p} \frac{\mathcal{Y}(s) + \mathcal{Y}(t)}{2} \left[\mathcal{Z}\_{s^{+}}^{\mathbf{a}}\ \mathfrak{C}(t) + \mathcal{Z}\_{t^{-}}^{\mathbf{a}}\ \mathfrak{C}(s)\right]$$

*If one takes Y*∗(*ω*) = *Y* ∗ (*ω*) *with* a = 1, *then from (13) and (18), we achieve the classical* H*–*H *Fejér inequality (see [26])*.

#### **4. Conclusions**

In applied sciences, convex functions and fractional calculus are essential. The new interval-valued left and right convex functions are presented in this article. Some novel Riemann–Liouville fractional integral H–H and Fejér-type inequalities are provided, utilizing the idea of interval-valued left and right convex functions and some supplementary interval analysis findings. Our results are a generalization of a number of previously published findings. In the future, we will use generalized interval and fuzzy Riemann–Liouville fractional operators to investigate this concept for generalized left and right convex *I*·*V-Fs* and *F-I*·*V-Fs* by using interval Katugampola fractional integrals and fuzzy Katugampola fractional integrals. For applications, see [53–56].

**Author Contributions:** Conceptualization, M.B.K.; methodology, M.B.K.; validation, S.T., H.G.Z., and M.S.S.; formal analysis, G.S.-G.; investigation, M.S.S.; resources, S.T. and J.E.M.-D.; data curation, H.G.Z.; writing—original draft preparation, H.G.Z., M.B.K., and G.S.-G.; writing—review and editing, S.T. and M.B.K.; visualization, J.E.M.-D., and H.G.Z.; supervision, M.S.S. and M.B.K.; project administration, M.B.K.; funding acquisition, M.S.S., G.S.-G., and H.G.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research. This work was funded by Taif University Researchers Supporting Project (number TURSP-2020/345), Taif University, Taif, Saudi Arabia. Moreover, the work of Santos-García was also partially supported by the Spanish project TRACES TIN2015-67522-C3-3-R.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Some Fuzzy Riemann–Liouville Fractional Integral Inequalities for Preinvex Fuzzy Interval-Valued Functions**

**Muhammad Bilal Khan 1 , Hatim Ghazi Zaini 2 , Jorge E. Macías-Díaz 3,4,\* , Savin Treant,aˇ 5,\* and Mohamed S. Soliman 6**

**Abstract:** The main objective of this study is to introduce new versions of fractional integral inequalities in fuzzy fractional calculus utilizing the introduced preinvexity. Due to the behavior of its definition, the idea of preinvexity plays a significant role in the subject of inequalities. The concepts of preinvexity and symmetry have a tight connection thanks to the significant correlation that has developed between both in recent years. In this study, we attain the Hermite-Hadamard (*H*·*H*) and Hermite-Hadamard-Fejér (*H*·*H* Fejér) type inequalities for preinvex fuzzy-interval-valued functions (preinvex *F*·*I*·*V*·*Fs*) via Condition C and fuzzy Riemann–Liouville fractional integrals. Furthermore, we establish some refinements of fuzzy fractional *H*·*H* type inequality. There are also some specific examples of the reported results for various preinvex functions deduced. To support the newly introduced ideal, we have provided some nontrivial and logical examples. The results presented in this research are a significant improvement over earlier results. This paper's awe-inspiring notions and formidable tools may energize and revitalize future research on this worthwhile and fascinating topic.

**Keywords:** preinvex fuzzy interval-valued function; fuzzy fractional integral operator; Hermite-Hadamard type inequality; Hermite-Hadamard Fejér type inequality

#### **1. Introduction**

Convex function theory has a wide range of potential applications in a variety of unique and fascinating disciplines of study. Furthermore, this theory is useful in a variety of fields, including physics, information theory, coding theory, engineering, optimization, and inequality theory. This theory is currently making a significant contribution to the extensions and improvements of a wide range of mathematical and practical fields. Many authors analyzed, celebrated, and executed their work on the concept of convexity, and used fruitful methodologies and novel ideas to extend its many variations in helpful ways. In the literature, several new families of classical convex functions have been proposed. The references [1–5] are provided for the benefit of the readers. Many authors and scientists have always attempted to contribute to the theory of inequality by producing high-quality work. Integral inequalities on convex functions, both derivative and integration, have likewise been a hot and engaging area of study in recent years. The theory of inequalities has significant applications in the field of applied analysis, such as geometric function theory, impulsive diffusion equations, coding theory, numerical analysis, and fractional

**Citation:** Khan, M.B.; Zaini, H.G.; Macías-Díaz, J.E.; Treant,a, S.; ˇ Soliman, M.S. Some Fuzzy Riemann–Liouville Fractional Integral Inequalities for Preinvex Fuzzy Interval-Valued Functions. *Symmetry* **2022**, *14*, 313. https:// doi.org/10.3390/sym14020313

Academic Editor: Alexander Zaslavski

Received: 4 January 2022 Accepted: 21 January 2022 Published: 3 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

calculus, to name a few. Sun [6] and co-workers [7] recently used the local fractional integral operator to generalize the Hermite-Hadamard condition for harmonically convex and s-preinvex functions. The references [8–13] are provided for the benefit of the readers.

Several writers have recently proposed novel inequalities for various types of convexities, preinvexities, statistical theory, and other topics. Several discussions show a tight connection between inequality theory and convex functions. Hanson examined the invex function in the context of bi-function *ϕ*(., .) for the first time in 1981 (see [14]). Following Hanson's work, Ben-Israel and Mond attempted to delve deeper into linked invexity, introducing the concepts of invex sets and preinvex functions for the first time (see [15]). Under certain conditions, the preinvex and invex functions in the form of differentiability are comparable, according to Mohan and Neogy [16]. Antczak [17] discovered and analyzed the features of preinvex functions for the first time in 2005.

Note that fuzzy mappings (*F*·*Ms*) are fuzzy-interval-valued functions. On the other hand, the concept of convex *F*·*Ms* from R*n* to the set of fuzzy numbers was introduced by Nanda and Kar [18], Syau [19], and Furukawa [20]. They also explored Lipschitz continuity of fuzzy valued mappings and created other types of convex *F*·*Ms*, such as logarithmic convex *F*·*Ms* and quasi-convex *F*·*Ms*. Based on Goetschel and Voxman's concept of ordering [21], Yan and Xu [22] introduced the conceptions of epigraphs and convexity of *F*·*Ms*, as well as the properties of convex *F*·*Ms* and quasi-convex *F*·*Ms*. Khan et al. [23–26] extended the class of convex *F*·*Ms* and defined *h*-convex and (*h*1, *h*2)-convex *F*·*I*·*V*·*Fs* using fuzzy partial order relation. Moreover, they introduced *H*·*H, H*·*H* Fejér, *H*·*H* fractional, *H*·*H* fractional Fejér for *h*-convex and (*h*1, *h*2)-convex *F*·*I*·*V*·*Fs* via fuzzy Riemannian and fuzzy Riemann–Liouville fractional integrals. Noor [27] proposed and investigated the notion of fuzzy preinvex mapping on the invex set. He also showed how to express the fuzzy optimality conditions of differentiable preinvex fuzzy mappings using variational inequalities. Recently Khan et al. [28] generalized the concept of preinvex fuzzy mappings in terms of (*h*1, *h*2)-preinvex *F*·*I*·*V*·*Fs.* Moreover, they established relation between *H*·*H* inequalities and (*h*1, *h*2)-preinvex *F*·*I*·*V*·*Fs* by using fuzzy Riemannian integrals. Recently Khan et al. [29–33] proposed the concepts of strongly preinvex *F*·*I*·*V*·*Fs,* higher strongly preinvex *F*·*I*·*V*·*Fs,* generalized strongly preinvex *F*·*I*·*V*·*Fs* and characterized their optimality conditions by introducing different variational like inequalities. Moreover, they proposed *H*·*H* inequalities for strongly preinvex *F*·*I*·*V*·*Fs* by utilizing fuzzy Riemannian.

At one step forward, Khan et al. introduced new classes of convex and generalized convex *F*·*I*·*V*·*Fs*, and derived new *H*·*H* type inequalities for log-s-convex *F*·*I*·*V*·*Fs* in the second sense [34], log-*h*-convex *F*·*I*·*V*·*Fs* [35] and the references therein. We refer to the readers for further analysis of literature on the applications and properties of fuzzy-interval, and inequalities and generalized convex *F*·*Ms*, see [36–56] and the references therein.

The goal of this study is to complete the fuzzy Riemann–Liouville fractional integrals for *F*·*I*·*V*·*Fs* and use these integrals to get the *H*·*H* inequalities. These integrals are also used to derive *H*·*H* type inequalities for preinvex *F*·*I*·*V*·*Fs*.

#### **2. Preliminaries**

Let K*C* be the space of all closed and bounded intervals of R and *η* ∈ K*C* be defined by

$$\eta = \left[ \eta\_{\*\prime} \,\,\eta^\* \right] = \left\{ \omega \in \mathbb{R} \, \middle| \,\eta\_\* \le \omega \le \eta^\* \right\}, \text{ } (\eta\_{\*\prime} \,\,\eta^\* \in \mathbb{R})$$

if *η*∗ = *η* ∗ then, *η* is said to be degenerate. In this article, all intervals will be non-degenerate intervals. If *η*∗ ≥ 0, then [*η*∗, *η* ∗ ] is called positive interval. The set of all positive interval is denoted by K + *C* and defined as K + *C* = {[*η*∗, *η* ∗ ] : [*η*∗, *η* ∗ ] ∈ K*C* and *η*∗ ≥ 0} .

Let *ς* ∈ R and *ςη* be defined by

$$
\xi \cdot \eta = \begin{cases}
\ \left[\xi \eta\_{\ast \prime} \ \xi \eta^{\ast}\right] \text{ if } \xi \ge 0, \\
\ \left[\xi \eta^{\ast} \ \xi \eta\_{\ast}\right] \text{ if } \xi < 0.
\end{cases}
\tag{1}
$$

Then the Minkowski difference *ξ* − *η*, addition *η* + *ξ* and *η* × *ξ* for *η*, *ξ* ∈ K*C* are defined by

$$\begin{array}{l} \left[ \mathfrak{f}\_{\ast \prime}^{\star} \mathfrak{f}^{\ast} \right] - \left[ \eta\_{\ast \prime} \eta^{\ast} \right] = \left[ \mathfrak{f}\_{\ast}^{\star} - \eta\_{\ast \prime} \ \mathfrak{f}^{\ast} - \eta^{\ast} \right] \, \prime \\\ \left[ \mathfrak{f}\_{\ast \prime}^{\star} \ \mathfrak{f}^{\ast} \right] + \left[ \eta\_{\ast \prime} \ \eta^{\ast} \right] = \left[ \mathfrak{f}\_{\ast}^{\star} + \eta\_{\ast \prime} \ \mathfrak{f}^{\ast} + \eta^{\ast} \right] \, \prime \end{array} \tag{2}$$

and

$$\begin{bmatrix} \tilde{\xi}^{\*} \ \tilde{\xi}^{\*} \end{bmatrix} \times \begin{bmatrix} \eta\_{\*} \ \eta^{\*} \end{bmatrix} = \begin{bmatrix} \min \{ \tilde{\xi}^{\*} \eta^{\*} \ \tilde{\xi}^{\*} \eta^{\*} \end{bmatrix} \begin{bmatrix} \xi^{\*} \eta^{\*} \end{bmatrix},\\ \max \{ \tilde{\xi}^{\*} \eta^{\*} \ \tilde{\xi}^{\*} \eta^{\*} \} \ \begin{array}{c} \max \{ \tilde{\xi}^{\*} \eta^{\*} \} \ \tilde{\xi}^{\*} \eta^{\*} \end{array} \begin{bmatrix} \tilde{\xi}^{\*} \eta^{\*} \end{bmatrix} \end{bmatrix}$$

The inclusion "⊆" means that

$$\xi \subseteq \eta \text{ if and only if, } [\xi\_{\*\*} \text{ } \xi^\*] \subseteq [\eta\_{\*\*} \text{ } \eta^\*], \text{ if and only if } \eta\_\* \le \xi\_{\*\*} \text{ } \xi^\* \le \eta^\* \tag{3}$$

**Remark 2.1.** [38] The relation " ≤*I* " defined on K*C* by

$$\left[\nabla\_{\ast \prime} \cdot \nabla^{\ast}\right] \le\_I \left[\eta\_{\ast \prime} \cdot \eta^{\ast}\right] \text{ if and only if } \nabla\_{\ast} \le \eta\_{\ast \prime} \nabla^{\ast} \le \eta^{\ast},\tag{4}$$

for all [∇∗, ∇∗ ] , [*η*∗, *η* ∗ ] ∈ K*C*, it is an order relation. For given [∇∗, ∇∗ ] , [*η*∗, *η* ∗ ] ∈ K*C*, we say that [∇∗, ∇∗ ] ≤*I* [*η*∗, *η* ∗ ] if and only if ∇∗ ≤ *η*∗, ∇∗ ≤ *η* ∗ or ∇∗ ≤ *η*∗, ∇∗ < *η* ∗ .

A fuzzy subset *A* of R is characterize by a mapping *ζ* : R → [0, 1] called the membership function, for each fuzzy set and *θ* ∈ (0, 1], then *θ*-level sets of *ζ* is denoted and defined as follows *ζθ* = {*u* ∈ R| *ζ*(*u*) ≥ *θ*}. If *θ* = 0, then *supp*(*ζ*) = {*ω* ∈ R | *ζ*(*ω*)i0} is called support of *ζ*. By [*ζ*] 0 we define the closure of *supp*(*ψ*).

Let F(R) be the family of all fuzzy sets and *ζ* ∈ F(R) denote the family of all nonempty sets. *ζ* ∈ F(R) be a fuzzy set. Then we define the following:

A fuzzy set is called a fuzzy number or fuzzy interval if it has properties (1), (2), (3) and (4). We denote by F0 the family of all intervals.

Let *ζ* ∈ F0 be a fuzzy-interval, if and only if, *θ*-levels [*ζ*] *θ* is a nonempty compact convex set of R. From these definitions, we have

$$\left[\zeta\right]^\theta = \left[\zeta\_\*(\theta) \text{ , } \zeta^\*(\theta)\right] \text{ .}$$

where

$$\mathcal{L}\_\*(\theta) = \inf \{ \omega \in \mathbb{R} \, | \, \zeta(\omega) \ge \theta \}, \\ \zeta^\*(\theta) = \sup \{ \omega \in \mathbb{R} \, | \, \zeta(\omega) \ge \theta \}. \tag{5}$$

**Proposition 2.2.** [47] If *ζ*, *η* ∈ F0 then relation " 4 " defined on F0 by

$$\mathcal{L} \preccurlyeq \eta \text{ if and only if, } \left[\zeta\right]^{\theta} \le\_I \left[\eta\right]^{\theta}, \text{ for all } \theta \in \left[0, 1\right] \text{ }\tag{6}$$

this relation is known as partial order relation.

For *ζ*, *η* ∈ F0 and *ς* ∈ R, the sum *ζ*+e *η*, product *ζ*×e *η*, scalar product *ς*.*ζ* and sum with scalar are defined by:

Then, for all *θ* ∈ [0, 1] , we have

$$\left[\left[\widetilde{\zeta} \dot{+} \eta\right]^{\theta} = \left[\widetilde{\zeta}\right]^{\theta} + \left[\eta\right]^{\theta}\right.\tag{7}$$

$$\left[\left[\widetilde{\mathbb{J}} \widetilde{\times} \eta\right]^{\theta} = \left[\widetilde{\mathbb{J}}\right]^{\theta} \times \left[\left.\eta\right]^{\theta}\right] \tag{8}$$

$$\left[\emptyset,\mathbb{Q}\right]^\theta = \emptyset . \left[\mathbb{Q}\right]^\theta. \tag{9}$$

$$\left[\mathfrak{J}\left[\mathfrak{J}+\mathfrak{J}\right]^{\theta}=\mathfrak{g}+\left[\mathfrak{J}\right]^{\theta}.\tag{10}$$

For *ψ* ∈ F0 such that *ζ* = *η*+e*ψ*, then by this result we have existence of Hukuhara difference of *ζ* and *η*, and we say that *ψ* is the H-difference of *ζ* and *η*, and denoted by *ζ*−e *η*. If H-difference exists, then

$$(\left(\psi\right)^{\*}\left(\theta\right) \;=\left(\zeta\overset{\sim}{-}\eta\right)^{\*}\left(\theta\right) \;=\zeta^{\*}\left(\theta\right) \;-\eta^{\*}\left(\theta\right),\\(\psi)\_{\*}\left(\theta\right) \;=\left(\zeta\overset{\sim}{-}\eta\right)\_{\*}\left(\theta\right) \;=\zeta\_{\*}\left(\theta\right) \;-\eta\_{\*}\left(\theta\right).$$

**Definition 2.3.** [36] A fuzzy map Ψ : [*u*, *ν*] ⊂ R → F0 is called *F*·*I*·*V*·*F*. For each *θ* ∈ [0, 1] , whose *θ*-levels define the family of *I*·*V*·*F* Ψ*θ* : [*u*, *ν*] ⊂ R → K*C* are given by Ψ*θ* (*ω*) = [Ψ∗(*ω*, *θ*) , Ψ∗ (*ω*, *θ*)] for all *ω* ∈ [*u*, *ν*] . Here, for each *θ* ∈ [0, 1] , the left and right real valued functions Ψ∗(*ω*, *θ*) , Ψ∗ (*ω*, *θ*) : [*u*, *ν*] → R are also called lower and upper functions of Ψ.

**Remark 2.4.** If Ψ : [*u*, *ν*] ⊂ R → F0 is a *F*·*I*·*V*·*F*, then Ψ(*ω*) is called continuous function at *ω* ∈ [*u*, *ν*] , if for each *θ* ∈ [0, 1] , both left and right real valued functions Ψ∗(*ω*, *θ*) and Ψ∗ (*ω*, *θ*) are continuous at *ω* ∈ [*u*, *ν*] .

The following FI Riemann–Liouville fractional integral operators were introduced by Allahviranloo et al. [40]:

**Definition 2.5.** Let *β* > 0 and *L*([*µ*, *υ*] , F0) be the collection of all Lebesgue measurable *F*·*I*·*V*·*Fs* on [*µ*, *υ*]. Then the fuzzy left and right Riemann–Liouville fractional integral of Ψ ∈ *L*([*µ*, *υ*] , F0) with order *β* > 0 are defined by

$$\mathcal{T}\_{\mu^{+}}^{\beta}\,\Psi(\omega) \,= \frac{1}{\Gamma(\beta)} \int\_{\mu}^{\omega} (\omega - \underline{\varsigma})^{\beta - 1} \Psi(\underline{\varsigma}) d\underline{\varsigma} \,\,\, (\omega > \mu) \tag{11}$$

and

$$\mathcal{T}\_{v^{-}}^{\mathfrak{f}}\left.\Psi(\omega)\right. = \frac{1}{\Gamma(\mathfrak{f})} \int\_{\omega}^{v} (\mathfrak{f}-\omega)^{\mathfrak{f}-1} \Psi(\mathfrak{f}) d\mathfrak{g}\,\,\left(\omega$$

respectively, where Γ(*ω*) = R ∞ 0 *ς ω*−1 *e* −*ςdς* is the Euler gamma function. The fuzzy left and right Riemann–Liouville fractional integral *ω* based on left and right end point functions can be defined, that is

$$\begin{aligned} \left[ \mathcal{Z}\_{\mu^{+}}^{\theta} \, \Psi(\omega) \right]^{\theta} &= \frac{1}{\Gamma(\theta)} \int\_{\mu}^{\omega} (\omega - \varsigma)^{\theta - 1} \Psi\_{\theta}(\varsigma) d\varsigma \\\\ \zeta &= \frac{1}{\Gamma(\theta)} \int\_{\mu}^{\omega} (\omega - \varsigma)^{\theta - 1} [\Psi\_{\ast}(\varsigma, \theta) \, \, \Psi^{\ast}(\varsigma, \theta)] d\varsigma \, \, (\omega > \mu) \end{aligned} \tag{13}$$

where

$$\mathcal{T}\_{\mu^{+}}^{\mathfrak{f}}\,^{\Psi}\_{\*}(\omega,\theta) \,= \frac{1}{\Gamma(\mathfrak{f})} \int\_{\mu}^{\omega} (\omega - \mathfrak{g})^{\mathfrak{f}-1} \Psi\_{\*}(\mathfrak{g},\theta) d\mathfrak{g} \,\,(\omega > \mu)\,\,\,\tag{14}$$

and

$$\mathcal{T}^{\delta}\_{\mu^{+}}\,\Psi^{\*}(\omega,\theta) \,=\frac{1}{\Gamma(\beta)}\int\_{\mu}^{\omega}(\omega-\emptyset)^{\beta-1}\Psi^{\*}(\emptyset,\theta)d\xi\,\,(\omega>\mu)\,\,\tag{15}$$

Similarly, the left and right end point functions can be used to define the right Riemann– Liouville fractional integral Ψ of *ω*.

**Definition 2.6.** [18]. The *F*·*I*·*V*·*F* Ψ : [*u*, *ν*] → F0 is called convex *F*·*I*·*V*·*F* on [*u*, *ν*] if

$$\Psi(\emptyset \omega + (1 - \emptyset)y) \nrightarrow \emptyset \Psi(\omega) \tilde{+} (1 - \emptyset) \Psi(y) \; \; \; \tag{16}$$

for all *ω*, *y* ∈ [*u*, *ν*] , *ς* ∈ [0, 1] , where for all Ψ(*ω*) < e0 for all *ω* ∈ [*u*, *ν*] . If (16) is reversed, then Ψ is called concave *F*·*I*·*V*·*F* on [*u*, *ν*]. Ψ is affine if and only if, it is both convex and concave *F*·*I*·*V*·*F*.

**Definition 2.7.** [27]. The *F*·*I*·*V*·*F* Ψ : [*u*, *ν*] → F0 is called preinvex *F*·*I*·*V*·*F* on invex interval [*u*, *ν*] if

$$\Psi(\omega + (1 - \varsigma)\varphi(\omega, y)) \preccurlyeq \xi \Psi(\omega) \bar{+} (1 - \varsigma)\Psi(y) \,, \tag{17}$$

for all *ω*, *y* ∈ [*u*, *ν*] , *ς* ∈ [0, 1] , where Ψ(*ω*) < e0 for all *ω* ∈ [*u*, *ν*] and *ϕ* : [*u*, *ν*] × [*u*, *ν*] → R. If (17) is reversed then, Ψ is called preconcave *F*·*I*·*V*·*F* on [*u*, *ν*]. Ψ is affine if and only if, it is both preinvex and preconcave *F*·*I*·*V*·*F*.

We need the following assumption regarding the function *ϕ* : [*u*, *ν*] × [*u*, *ν*] → R, which plays an important role in upcoming main results.

#### **Condition C.** [16]

$$
\begin{aligned}
\label{eq:center}
\varphi(y\_\prime\omega + \tau\varphi(y\_\prime\omega)) &= (1-\tau)\varphi(y\_\prime\omega) \\
\label{eq:center}
\varphi(\omega,\omega + \tau\varphi(y\_\prime\omega)) &= -\tau\varphi(y\_\prime\omega)
\end{aligned}
$$

Note that ∀ *ω*, *y* ∈ [*u*, *ν*] and *ς* ∈ [0, 1], then from Condition C we have

$$
\varphi(\omega + \tau\_2 \varphi(y, \omega), \omega + \tau\_1 \varphi(y, \omega)) = (\tau\_2 - \tau\_1)\varphi(y, \omega).
$$

Clearly for *τ* = 0, we have *ξ*(*y*, *ω*) = 0 if and only if *y* = *ω*, for all *ω*, *y* ∈ [*u*, *ν*]. For the application of Condition C, see [27–33].

**Theorem 2.8.** [28] *Let* [*u*, *ν*] *be an invex set with resoect to bifunvtion ϕ and* Ψ : [*u*, *ν*] → F*C*(R) *be a F*·*I*·*V*·*F with* Ψ(*ω*) < e0*, whose θ-levels define the family of I*·*V*·*Fs* Ψ*θ* : [*u*, *ν*] ⊂ R → K*C* + *are given by*

$$\Psi\_{\theta}(\omega) = \left[ \Psi\_{\*}(\omega, \theta) \; , \; \Psi^{\*}(\omega, \theta) \right] \; , \; \forall \; \omega \in \left[ \mathfrak{u} \; , \nu \right] \tag{18}$$

*for all ω* ∈ [*u*, *ν*] *and for all θ* ∈ [0, 1]*. Then,* Ψ *is preinvex* F·I·V·F *on* [*u*, *ν*] , *if and only if, for all θ* ∈ [0, 1] , Ψ∗(*ω*, *θ*) *and* Ψ∗ (*ω*, *θ*) *both are preinvex functions.*

**Remark 2.9.** If *ϕ*(*ω*, *y*) = *ω* − *y*, then we obtain inequality (16).

If Ψ∗(*ω*, *θ*) = Ψ∗ (*ω*, *θ*) with *θ* = 1, then from (17), we obtain the definition of classical preinvex function, see [16].

If Ψ∗(*ω*, *θ*) = Ψ∗ (*ω*, *θ*) with *ϕ*(*ω*, *y*) = *ω* − *y* and *θ* = 1, then from (17), we obtain the definition of classical convex function.

#### **3. Fuzzy-Interval Fractional Hermite-Hadamard Inequalities**

The major goal of this section is to build a new version of fractional *H*·*H* and *H*·*H* Fejér type inequality in the mode of preinvex *F*·*I*·*V*·*Fs*, which is a classical studied topic. We also study some related inequalities. In what follows, we denote by *L*([*u*, *u* + *ϕ*(*ν*, *u*)] , F0) the family of Lebesgue measureable *F*·*I*·*V*·*Fs*.

**Theorem 3.1.** *Let* Ψ : [*u*, *u* + *ϕ*(*ν*, *u*)] → F0 *be a preinvex F*·*I*·*V*·*F on* [*u*, *u* + *ϕ*(*ν*, *u*)] , *whose θ-levels define the family of I*·*V*·*Fs* Ψ*θ* : [*u*, *u* + *ϕ*(*ν*, *u*)] ⊂ R → K*C* + *are given by* Ψ*θ* (*ω*) = [Ψ∗(*ω*, *θ*) , Ψ∗ (*ω*, *θ*)] *for all ω* ∈ [*u*, *u* + *ϕ*(*ν*, *u*)] *and for all θ* ∈ [0, 1]*. If ϕ satisfies Condition C and* Ψ ∈ *L*([*u*, *u* + *ϕ*(*ν*, *u*)] , F0)*, then*

$$\mathbb{P}\left(\frac{2u+\varrho(\nu,u)}{2}\right) \preccurlyeq \frac{\Gamma(\beta+1)}{2(\varrho(\nu,u))^{\beta}} \Big[T^{\beta}\_{u^{+}} \, \mathbb{P}(u+\varrho(\nu,u)) \, \overline{+} T^{\beta}\_{u+\varrho(\nu,u)^{-}} \, \mathbb{P}(u)\Big] \prec \frac{\mathbb{P}(u)\overline{+} \Psi(u+\varrho(\nu,u))}{2} \preccurlyeq \frac{\Psi(u)\overline{+} \Psi(v)}{2} \tag{19}$$

If Ψ(*ω*) is preconcave *F*·*I*·*V*·*F* then

$$\mathbb{P}\left(\frac{2u+\varrho(\nu,u)}{2}\right) \xleftarrow{\Gamma(\mathfrak{F}+1)} \frac{\Gamma(\mathfrak{F}+1)}{2(\varrho(\nu,u))^{\mathfrak{F}}} \left[\mathbb{Z}\_{u^{+}}^{\mathfrak{G}}\,\Psi(u+\varrho(\nu,u))\,\overline{+}\mathbb{Z}\_{u+\varrho(\nu,u)^{-}}^{\mathfrak{G}}\,\Psi(u)\right] \xleftarrow{\Psi(u)} \frac{\Psi(u)\,\overline{+}\Psi(u,u)}{2} \succeq \frac{\Psi(u)\,\overline{+}\Psi(\nu)}{2} \tag{20}$$

**Proof.** Let Ψ : [*u*, *u* + *ϕ*(*ν*, *u*)] → F0 be a preinvex *F*·*I*·*V*·*F*. If Condition C holds then, by hypothesis, we have that

$$2\Psi\left(\frac{2\iota+\varrho(\nu,\iota)}{2}\right) \preccurlyeq \Psi(\iota+(1-\varrho)\varrho(\nu,\iota)) \cdot \widetilde{+}\Psi(\iota+\xi\varrho(\nu,\iota))$$

Therefore, for every *θ* [0, 1], we have

$$2\mathbb{1}\_\*\left(\frac{2u+\varrho(\nu,\mu)}{2},\theta\right) \le \mathbb{1}\_\*(u+\left(1-\varrho\right)\varrho(\nu,u),\theta) + \mathbb{1}\_\*(u+\xi\varrho(\nu,u),\theta),$$

$$2\mathbb{1}^\*\left(\frac{2u+\varrho(\nu,u)}{2},\theta\right) \le \mathbb{1}^\*(u+\left(1-\varrho\right)\varrho(\nu,u),\theta) + \mathbb{1}^\*(u+\xi\varrho(\nu,u),\theta).$$

Multiplying both sides by *ς β*−1 and integrating the obtained result with respect to *ς* over (0, 1), we have

$$2\int\_0^1 \xi^{\beta - 1} \Psi\_\*\left(\frac{2u + \varrho(\nu, \mu)}{2}, \theta\right) d\xi$$

$$\leq \int\_0^1 \xi^{\beta - 1} \Psi\_\*\left(u + \left(1 - \xi\right)\varrho(\nu, u\right), \theta\right) d\xi + \int\_0^1 \xi^{\beta - 1} \Psi\_\*\left(u + \xi\varrho(\nu, u), \theta\right) d\xi.$$

$$2\int\_0^1 \xi^{\beta - 1} \Psi^\*\left(\frac{2u + \varrho(\nu, u)}{2}, \theta\right) d\xi$$

$$\leq \int\_0^1 \xi^{\beta - 1} \Psi^\*\left(u + \left(1 - \xi\right)\varrho(\nu, u), \theta\right) d\xi + \int\_0^1 \xi^{\beta - 1} \Psi^\*\left(u + \xi\varrho(\nu, u), \theta\right) d\xi.$$

Let *ω* = *u* + (1 − *ς*)*ϕ*(*ν*, *u*) and *y* = *u* + *ςϕ*(*ν*, *u*) . Then we have

2 *β*Ψ∗ 2*u*+*ϕ*(*ν*,*u*) 2 , *θ* ≤ 1 (*ϕ*(*ν*,*u*))*β u*+*ϕ*(*ν*,*u*) R *u* (*u* + *ϕ*(*ν*, *u*) − *y*) *β*−1Ψ∗(*y*, *θ*)*dy* + 1 (*ϕ*(*ν*,*u*))*β u*+*ϕ*(*ν*,*u*) R *u* (*ω* − *u*) *β*−1Ψ∗(*ω*, *θ*)*dω* 2 *β*Ψ∗ 2*u*+*ϕ*(*ν*,*u*) 2 , *θ* ≤ 1 (*ϕ*(*ν*,*u*))*β u*+*ϕ*(*ν*,*u*) R *u* (*u* + *ϕ*(*ν*, *u*) − *y*) *β*−1Ψ∗ (*y*, *θ*)*dy* + 1 (*ϕ*(*ν*,*u*))*β u*+*ϕ*(*ν*,*u*) R *u* (*ω* − *u*) *β*−1Ψ∗ (*ω*, *θ*)*dω*, ≤ Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗(*u*, *θ*) ≤ Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗ (*u*, *θ*) ,

That is

$$\begin{aligned} \frac{2}{\beta} \Big[ \mathbf{V}\_{\ast} \left( \frac{2u + q(\nu \mu)}{2}, \theta \right) \Big. \left. \mathbf{V}^{\ast} \left( \frac{2u + q(\nu \mu)}{2}, \theta \right) \right] \\\\ \leq \underset{\left(\varphi(\nu, u)\right)^{\beta}}{\Gamma(\varphi(\nu, u))^{\beta}} \Big[ \mathcal{Z}\_{\mu^{+}}^{\theta} \left( \boldsymbol{\Psi}\_{\ast} \left( \boldsymbol{u} + \boldsymbol{\varrho}(\nu, u) \right), \theta \right) + \mathcal{Z}\_{\boldsymbol{u} + \boldsymbol{\varrho}(\nu, \mu)^{-}}^{\theta} \left. \boldsymbol{\Psi}\_{\ast} \left( \boldsymbol{u} + \boldsymbol{\varrho}(\nu, u) \right), \theta \right) \\\\ \leq \underset{\left(\varphi(\nu, u)\right)^{\beta}}{\Gamma(\varphi(\nu, u))^{\beta}} \left[ \boldsymbol{\Psi}\_{\ast} \left( \boldsymbol{u} + \boldsymbol{\varrho}(\nu, u) \right), \theta \right] + \mathcal{Z}\_{\boldsymbol{u} - \boldsymbol{\varrho}}^{\theta} \left( \boldsymbol{u} + \boldsymbol{\varrho}(\nu, u) \right), \theta \right] \end{aligned}$$

Thus,

$$\frac{2}{\beta} \,\, \Psi \left( \frac{2u + \varrho(\nu, u)}{2} \right) \,\, \preccurlyeq \frac{\Gamma(\beta)}{(\varrho(\nu, u))^{\beta}} \left[ \mathcal{T}\_{u^{+}}^{\beta} \,\, \Psi(u + \varrho(\nu, u)) \,\widetilde{+} \mathcal{T}\_{u + \varrho(\nu, u)^{-}}^{\beta} \,\, \Psi(u) \right] \tag{21}$$

In a similar way as above, we have

$$\frac{\Gamma(\beta)}{\left(\varphi(\upsilon,u)\right)^{\beta}} \left[ \mathcal{Z}\_{u^{+}}^{\theta} \cdot \Psi(u+\varrho(\upsilon,u)) \tilde{+} \mathcal{Z}\_{u+\varrho(\upsilon,u)}^{\theta} \cdot \Psi(u) \right] \prec \frac{\Psi(u) \tilde{+} \Psi(u+\varrho(\upsilon,u))}{2} \prec \frac{\Psi(u) \tilde{+} \Psi(\upsilon)}{2}.\tag{22}$$

Combining (21) and (22), we have

$$\mathbb{P}\left(\frac{2u+\varrho(\nu,u)}{2}\right) \preccurlyeq \frac{\Gamma(\beta+1)}{2(\varrho(\nu,u))^{\beta}} \left[\mathcal{Z}\_{u^{+}}^{\beta}\,\,\Psi(u+\varrho(\nu,u))\widetilde{+}\mathcal{Z}\_{u+\varrho(\nu,u)^{-}}^{\beta}\,\,\Psi(u)\right] \\ \preccurlyeq \frac{\Psi(u)\widetilde{+}\Psi(u+\varrho(\nu,u))}{2} \preccurlyeq \frac{\Psi(u)\widetilde{+}\Psi(\nu,u)}{2} \preccurlyeq \frac{\Psi(u)\widetilde{+}\Psi(\nu,u)}{2} \preccurlyeq \frac{\Psi(u)\widetilde{+}\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u)\Psi(u)}{2} \preccurlyeq \frac{\Psi(u)\Psi(u)\Psi(u}$$

Hence, the required result.

#### **Remark 3.2.** From Theorem 3.1 we clearly see that

If *ϕ*(*ω*, *y*) = *ω* − *y*, then from Theorem 3.1, we get following result in fuzzy fractional calculus, see [23].

$$\Psi\left(\frac{\mu+\nu}{2}\right) \preccurlyeq \frac{\Gamma(\beta+1)}{2(\nu-\mu)^{\beta}} \left[ \mathcal{I}\_{\mu^{+}}^{\beta} \,\,\Psi(\nu) \widetilde{+} \mathcal{I}\_{\nu^{-}}^{\beta} \,\,\Psi(\mu) \right] \preccurlyeq \frac{\Psi(\mu) \widetilde{+} \Psi(\nu)}{2}.$$

Let *β* = 1. Then Theorem 3.1 reduces to the result for preinvex *F*·*I*·*V*·*F* given in [28]:

$$\Psi\left(\frac{2u+\varrho(\nu,\mu)}{2}\right) \preccurlyeq \frac{1}{\varrho(\nu,\mu)} \int\_{u}^{u+\varrho(\nu,\mu)} \Psi(\omega)d\omega \preccurlyeq \frac{\Psi(u)\upharpoonright \Psi(\nu)}{2}.$$

Let *β* = 1 and *ϕ*(*ω*, *y*) = *ω* − *y*. Then Theorem 3.1 reduces to the result for convex *F*·*I*·*V*·*F* given in [26]:

$$\,\_2\Psi\left(\frac{\iota+\nu}{2}\right) \prec \frac{1}{\nu-\iota} \int\_{\iota}^{\nu} \Psi(\omega)d\omega \prec \frac{\Psi(\iota) \preccurlyeq \Psi(\nu)}{2}$$

Let *β* = 1 = *θ* and Ψ∗(*ω*, *θ*) = Ψ∗ (*ω*, *θ*) with *ϕ*(*ω*, *y*) = *ω* − *y*. Then from Theorem 3.1 we obtain classical *H*·*H* Fejér type inequality.

**Example 3.3.** Let *β* = 1 2 , *ω* ∈ [2, 2 + *ϕ*(3, 2)], and the *F*·*I*·*V*·*F* Ψ : [*u*, *u* + *ϕ*(*ν*, *u*)] = [2, 2 + *ϕ*(3, 2)] → F0, defined by

$$\Psi(\omega)(\theta) = \begin{cases} \frac{\theta}{2-\omega^{\frac{1}{2}}} & \theta \in \left[0, 2-\omega^{\frac{1}{2}}\right] \\\\ \frac{2\left(2-\omega^{\frac{1}{2}}\right)-\theta}{2-\omega^{\frac{1}{2}}} & \theta \in \left(2-\omega^{\frac{1}{2}}, 2\left(2-\omega^{\frac{1}{2}}\right)\right] \\\\ 0 & \theta \text{ otherwise} \end{cases}$$

Then, for each *θ* ∈ [0, 1] , we have Ψ*θ* (*ω*) = h *θ* 2 − *ω* 1 2 , (2 − *θ*) 2 − *ω* 1 2 i. Since left and right end point functions Ψ∗(*ω*, *θ*) = *θ* 2 − *ω* 1 2 , Ψ∗ (*ω*, *θ*) = (2 − *θ*) 2 − *ω* 1 2 , are preinvex functions with respect to *ϕ*(*ν*, *u*) = *ν* − *u*, for each *θ* ∈ [0, 1], then Ψ(*ω*) is preinvex *F*·*I*·*V*·*F*. We clearly see that Ψ ∈ *L*([*u*, *u* + *ϕ*(*ν*, *u*)] , F0) and

$$\mathbb{P}\_\*\left(\frac{2\mu + \varrho(\nu,\mu)}{2}, \theta\right) = \mathbb{P}\_\*\left(\frac{5}{2}, \theta\right) \\ = \theta \frac{4 - \sqrt{10}}{2}$$

$$\Psi^\*\left(\frac{2u+\varrho(\nu,u)}{2},\theta\right) = \Psi^\*\left(\frac{5}{2},\theta\right) \\ = (2-\theta)\frac{4-\sqrt{10}}{2}$$

$$\frac{\Psi\_\*(u,\theta) + \Psi\_\*(u+\varrho(\nu,u),\theta)}{2} = \theta\left(\frac{4-\sqrt{2}-\sqrt{3}}{2}\right)$$

$$\frac{\Psi^\*(u,\theta) + \Psi^\*(u+\varrho(\nu,u),\theta)}{2} = (2-\theta)\left(\frac{4-\sqrt{2}-\sqrt{3}}{2}\right)$$

Note that

Γ(*β* + 1) 2(*ϕ*(*ν*, *u*))*β* I *β u*+ Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗(*u*, *θ*) = Γ 3 2 2 1 √ *π* 2+*ϕ*(3,2) Z 2 (3 − *ω*) −1 2 . *θ* 2 − *ω* 1 2 *dω* + Γ 3 2 2 1 √ *π* 2+*ϕ*(3,2) Z 2 (*ω* − 2) −1 2 . *θ* 2 − *ω* 1 2 *dω* = 1 4 *θ* 7393 10, 000 + 9501 10, 000 = *θ* 8447 20, 000 Γ(*β* + 1) 2(*ϕ*(*ν*, *u*))*β* I *β u*+ Ψ ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) + I *β u*+*ϕ*(*ν*,*u*) − Ψ ∗ (*u*, *θ*) = Γ 3 2 2 1 √ *π* 2+*ϕ*(3,2) Z 2 (3 − *ω*) −1 2 . (2 − *θ*) 2 − *ω* 1 2 *dω* + Γ 3 2 2 1 √ *π* 2+*ϕ*(3,2) Z 2 (*ω* − 2) −1 2 . (2 − *θ*) 2 − *ω* 1 2 *dω* = 1 4 (2 − *θ*) 7393 10, 000 + 9501 10, 000 = (2 − *θ*) 8447 20, 000

Therefore

$$\left[\theta \frac{4-\sqrt{10}}{2}, (2-\theta)\frac{4-\sqrt{10}}{2}\right] \leq\_I \left[\theta \frac{8447}{20,000}, (2-\theta)\frac{8447}{20,000}\right] \leq\_I \left[\theta \left(\frac{4-\sqrt{2}-\sqrt{3}}{2}\right), (2-\theta)\left(\frac{4-\sqrt{2}+\sqrt{3}}{2}\right)\right]$$

and Theorem 3.1 is verified.

It is well known fact that *H*·*H* Fejér type inequality is a generalization of *H*·*H* type inequality. In Theorem 3.4 and Theorem 3.5, we obtain second and first fuzzy fractional *H*·*H* Fejér type inequalities for introduced preinvex *F*·*I*·*V*·*F.*

**Theorem 3.4.** *Let* Ψ : [*u*, *u* + *ϕ*(*ν*, *u*)] → F0 *be a preinvex F*·*I*·*V*·*F with u* < *ν, whose θlevels define the family of I*·*V*·*F* Ψ*θ* : [*u*, *u* + *ϕ*(*ν*, *u*)] ⊂ R → K*C* + *are given by* Ψ*θ* (*ω*) = [Ψ∗(*ω*, *θ*), Ψ∗ (*ω*, *θ*)] *for all ω* ∈ [*u*, *u* + *ϕ*(*ν*, *u*)] *and for all θ* ∈ [0, 1]*. Let* Ψ ∈

*L*([*u*, *u* + *ϕ*(*ν*, *u*)], F0) *and* Ω : [*u*, *u* + *ϕ*(*ν*, *u*)] → R, Ω(*ω*) ≥ 0, *symmetric with respect to* 2*u*+*ϕ*(*ν*,*u*) 2 . *If ϕ satisfies Condition C, then*

$$\begin{cases} \begin{aligned} &\mathcal{Z}\_{u^{+}}^{\tilde{\mathcal{P}}} \, \, \Psi \Omega(u + \varrho(\boldsymbol{\nu}, \boldsymbol{u})) \tilde{+} \mathcal{Z}\_{u + \varrho(\boldsymbol{\nu}, \boldsymbol{u})^{-}}^{\tilde{\mathcal{P}}} \, \Psi \Omega(u) \Big{]} \end{aligned} \\\\ \begin{aligned} &\mathcal{Z}\_{\boldsymbol{\nu}} \, \frac{\Psi(\boldsymbol{u}) \tilde{+} \Psi(\boldsymbol{u} + \varrho(\boldsymbol{\nu}, \boldsymbol{u}))}{2} \bigg[ \mathcal{Z}\_{u^{+}}^{\tilde{\mathcal{P}}} \, \Omega(\boldsymbol{u} + \varrho(\boldsymbol{\nu}, \boldsymbol{u})) \, \, \, \mathcal{Z}\_{u + \varrho(\boldsymbol{\nu}, \boldsymbol{u})^{-}}^{\tilde{\mathcal{P}}} \, \Omega(\boldsymbol{u}) \Big{]} \end{aligned} \\\\ &\prec \begin{aligned} \begin{aligned} \mathcal{X}(\boldsymbol{u}) \tilde{+} \Psi(\boldsymbol{\nu}) \Big{]} \, \mathcal{Z}\_{u^{+}}^{\tilde{\mathcal{P}}} \, \Omega(\boldsymbol{u} + \varrho(\boldsymbol{\nu}, \boldsymbol{u})) \, \, \widetilde{+} \mathcal{Z}\_{u + \varrho(\boldsymbol{\nu}, \boldsymbol{u})^{-}}^{\tilde{\mathcal{P}}} \, \Omega(\boldsymbol{u}) \end{aligned} \end{aligned} \end{cases} \end{cases} \end{cases} \tag{23}$$

If Ψ is preconcave *F*·*I*·*V*·*F*, then inequality (23) is reversed.

**Proof.** Let Ψ be a preinvex *F*·*I*·*V*·*F* and *ς β*−1Ω(*u* + (1 − *ς*)*ϕ*(*ν*, *u*)) ≥ 0. Then, for each *θ* ∈ [0, 1] , we have

$$\begin{split} \xi^{\delta-1} \Psi\_{\*} (\mathsf{u} + (1-\xi)\boldsymbol{\varrho}(\mathsf{v},\mathsf{u}) \, , \theta) \Omega(\mathsf{u} + (1-\xi)\boldsymbol{\varrho}(\mathsf{v},\mathsf{u})) \\ \leq \xi^{\delta-1} (\xi \mathsf{Y}\_{\*} (\mathsf{u},\theta) \, + \, (1-\xi)\boldsymbol{\Psi}\_{\*} (\mathsf{u} + \boldsymbol{\varrho}(\mathsf{v},\mathsf{u}) \, , \theta)) \Omega(\mathsf{u} + \, (1-\xi)\boldsymbol{\varrho}(\mathsf{u} + \boldsymbol{\varrho}(\mathsf{v},\mathsf{u}) \, , \mathsf{u})) \\ \leq \xi^{\delta-1} \boldsymbol{\Psi}^{\*} (\mathsf{u} + \, (1-\xi)\boldsymbol{\varrho}(\mathsf{v},\mathsf{u}) \, , \theta) \Omega(\mathsf{u} + \, (1-\xi)\boldsymbol{\varrho}(\mathsf{v},\mathsf{u})) \\ \leq \xi^{\delta-1} (\boldsymbol{\varrho}\boldsymbol{\Psi}^{\*} (\mathsf{u},\mathsf{u}) \, + \, (1-\xi)\boldsymbol{\Psi}^{\*} (\mathsf{u} + \boldsymbol{\varrho}(\mathsf{v},\mathsf{u}) \, , \theta)) \Omega(\mathsf{u} + \, (1-\xi)\boldsymbol{\varrho}(\mathsf{u} + \boldsymbol{\varrho}(\mathsf{v},\mathsf{u}) \, , \mathsf{u})) \,. \end{split} \tag{24}$$

and

*ς β*−1Ψ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*)Ω(*u* + *ςϕ*(*ν*, *u*)) ≤ *ς β*−1 ((1 − *ς*)Ψ∗(*u*, *θ*) + *ς*Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*))Ω(*u* + *ςϕ*(*ν*, *u*)) *ς β*−1Ψ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*)Ω(*u* + *ςϕ*(*ν*, *u*)) ≤ *ς β*−1 ((1 − *ς*)Ψ∗ (*u*, *θ*) + *ς*Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*))Ω(*u* + *ςϕ*(*ν*, *u*)) (25)

After adding (24) and (25), and integrating over [0, 1] , we get

$$\begin{split} &\int\_{0}^{1}\xi^{\beta-1}\Psi\_{\star}\left(u+(1-\xi)\varrho(\upsilon,u),\theta\right)\Omega(u+(1-\xi)\varrho(\upsilon,u))d\xi \\ &\qquad +\int\_{0}^{1}\xi^{\beta-1}\Psi\_{\star}\left(u+\xi\varrho(\upsilon,u),\theta\right)\Omega(u+\xi\varrho(\upsilon,u))d\xi \\ \leq &\int\_{0}^{1}\left[\begin{array}{cc} \xi^{\beta-1}\Psi\_{\star}\left(u,\theta\right)\{\xi\Omega(u+(1-\xi)\varrho(\upsilon,u))+(1-\xi)\Omega\left(u+\xi\varrho(\upsilon,u)\right)\} \\ +\xi^{\beta-1}\Psi\_{\star}\left(u+\varrho(\upsilon,u),\theta\right)\{(1-\xi)\Omega(u+(1-\xi)\varrho(\upsilon,u))+\xi\Omega(u+\xi\varrho(\upsilon,u))\} \end{array}\right] d\xi, \\ &\qquad\qquad +\int\_{0}^{1}\xi^{\beta-1}\Psi\_{\star}^{\ast}\left(u+(1-\xi)\varrho(\upsilon,u),\theta\right)\Omega(u+(1-\xi)\varrho(\upsilon,u))d\xi \\ &\qquad\qquad\qquad +\int\_{0}^{1}\xi^{\beta-1}\Psi\_{\star}^{\ast}\left(u+(1-\xi)\varrho(\upsilon,u)\right)\Omega(u+(1-\xi)\varrho(\upsilon,u))d\xi \\ \leq &\int\_{0}^{1}\left[\begin{array}{cc} \xi^{\beta-1}\Psi^{\ast}\left(u,\theta\right)\left\{\xi\Omega(u+(1-\xi)\varrho(\upsilon,u))+(1-\xi)\Omega(u+\xi\varrho(\upsilon,u))\right\} \\ +\xi^{\beta-1}\Psi^{\ast}\left(u+\varrho(\upsilon,u),\theta\right)\left\{(1-\xi)\varrho(\upsilon,u)\right\}+\xi\Omega(u+(1-\xi)\varrho(\upsilon,u))\right\} \end{array}\right] d\xi, \\ =$$

Since Ω is symmetric, then

= [Ψ∗(*u*, *θ*) + Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*)] R 1 0 *ς β*−1Ω(*u* + *ςϕ*(*ν*, *u*)) *dς* = [Ψ∗ (*u*, *θ*) + Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*)] R 1 0 *ς β*−1Ω(*u* + *ςϕ*(*ν*, *u*)) *dς*. = Ψ∗(*u*, *θ*) +Ψ∗(*u*+*ϕ*(*ν*,*u*) , *θ*) 2 Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ω(*u*) , = Ψ∗ (*u*, *θ*) +Ψ∗ (*u*+*ϕ*(*ν*,*u*) , *θ*) 2 Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ω(*u*) .  (27)

Then from (26), we have

Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗Ω(*u*) ≤ Ψ∗(*u*, *θ*) +Ψ∗(*u*+*ϕ*(*ν*,*u*) , *θ*) 2 Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ω(*u*) ≤ Ψ∗(*u*, *θ*) +Ψ∗(*u*+*ϕ*(*ν*,*u*) , *θ*) 2 Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ω(*u*) , Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗Ω(*u*) ≤ Ψ∗ (*u*, *θ*) +Ψ∗ (*u*+*ϕ*(*ν*,*u*) , *θ*) 2 Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ω(*u*) ≤ Ψ∗ (*u*, *θ*) +Ψ∗ (*u*+*ϕ*(*ν*) , *θ*) 2 Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ω(*u*) ,

that is

$$\frac{\Gamma(\boldsymbol{\beta})}{\Gamma(\boldsymbol{\rho}(\boldsymbol{\nu},\boldsymbol{\mu}))^{\boldsymbol{\beta}}} \left[ \left[ \boldsymbol{\mathcal{T}}\_{\boldsymbol{u}^{\boldsymbol{\beta}}}^{\boldsymbol{\beta}} \,\,\boldsymbol{\Psi}\_{\ast} \boldsymbol{\Omega}(\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})) \right. \\ \left. + \boldsymbol{\mathcal{T}}\_{\boldsymbol{u}+\boldsymbol{\rho}(\boldsymbol{\nu},\boldsymbol{\mu})}^{\boldsymbol{\beta}} \,\,\boldsymbol{\Psi}\_{\ast} \boldsymbol{\Omega}(\boldsymbol{u}) \right] \,, \,\boldsymbol{\mathcal{T}}\_{\boldsymbol{u}^{\boldsymbol{\beta}}}^{\boldsymbol{\beta}} \,\,\boldsymbol{\Psi}^{\ast} \boldsymbol{\Omega}(\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})) \right. \\ \left. + \boldsymbol{\mathcal{T}}\_{\boldsymbol{u}+\boldsymbol{\rho}(\boldsymbol{\nu},\boldsymbol{\mu})}^{\boldsymbol{\beta}} \,\,\boldsymbol{\Psi}\_{\ast} \boldsymbol{\Omega}(\boldsymbol{u}+\boldsymbol{\rho}(\boldsymbol{\nu},\boldsymbol{\mu}),\boldsymbol{\Theta}) \right] \,\,\boldsymbol{\Omega}(\boldsymbol{u}+\boldsymbol{\rho}(\boldsymbol{\nu},\boldsymbol{\mu})) $$

$$ \leq\_{I} \frac{\Gamma(\boldsymbol{\beta})}{(\boldsymbol{\rho}(\boldsymbol{\nu},\boldsymbol{\mu}))^{\boldsymbol{\beta}}} \left[ \frac{\mathsf{F}\_{\boldsymbol{u}}(\boldsymbol{\nu},\boldsymbol{\mu}) + \mathsf{F}\_{\boldsymbol{u}}(\boldsymbol{\nu},\boldsymbol{\mu})}{2} , \,\frac{\mathsf{F}^{\*}(\boldsymbol{u},\boldsymbol{\theta})}{2} \,\,\frac{\mathsf{F}^{\*}(\boldsymbol{u},\boldsymbol{\theta})}{2} \right] \,\Big[ \mathscr{T}\_{\boldsymbol{u}^{\boldsymbol{\beta}}}^{\boldsymbol{\beta}} \,\,\Omega(\boldsymbol{u}+\boldsymbol{\$$

hence

$$\begin{aligned} & \left[ \mathcal{I}\_{\boldsymbol{\mu}^+}^{\boldsymbol{\beta}} \, \Psi \Omega(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{\mu})) \, \xrightarrow{\sim} \mathcal{I}\_{\boldsymbol{\mu} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{\mu})^-}^{\boldsymbol{\beta}} \, \Psi \Omega(\boldsymbol{\mu}) \right] \\ \prec & \frac{\Psi(\boldsymbol{u}) \hat{+} \Psi(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{\mu}))}{2} \left[ \mathcal{I}\_{\boldsymbol{\mu}^+}^{\boldsymbol{\beta}} \, \Omega(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{\mu})) \, \right. + \mathcal{I}\_{\boldsymbol{\mu} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{\mu})^-}^{\boldsymbol{\beta}} \, \Omega(\boldsymbol{\mu}) \right] \\ \prec & \frac{\Psi(\boldsymbol{u}) \hat{+} \Psi(\boldsymbol{\nu})}{2} \left[ \mathcal{I}\_{\boldsymbol{\mu}^+}^{\boldsymbol{\beta}} \, \Omega(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{\mu})) \, \widetilde{+} \mathcal{I}\_{\boldsymbol{\mu} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{\mu})^-}^{\boldsymbol{\beta}} \, \Omega(\boldsymbol{u}) \right] \end{aligned}$$

**Theorem 3.5.** *Let* Ψ : [*u*, *u* + *ϕ*(*ν*, *u*)] → F0 *be a preinvex F*·*I*·*V*·*F with u* < *ν, whose θlevels define the family of I*·*V*·*Fs* Ψ*θ* : [*u*, *u* + *ϕ*(*ν*, *u*)] ⊂ R → K*C* + *are given by* Ψ*θ* (*ω*) = [Ψ∗(*ω*, *θ*) , Ψ∗ (*ω*, *θ*)] *for all ω* ∈ [*u*, *u* + *ϕ*(*ν*, *u*)] *and for all θ* ∈ [0, 1]*. If* Ψ ∈ *L* ([*u*, *u* + *ϕ*(*ν*, *u*)], F0) *and* Ω : [*u*, *u* + *ϕ*(*ν*, *u*)] → R, Ω(*ω*) ≥ 0, *symmetric with respect to* 2*u*+*ϕ*(*ν*,*u*) 2 *. If ϕ satisfies Condition C and then*

$$\begin{split} & \mathbb{1} \left( \frac{2u + \varrho(\nu, \boldsymbol{\mu})}{2} \right) \left[ \mathcal{T}\_{\boldsymbol{u}^{+}}^{\mathcal{G}} \, \Omega(\boldsymbol{u} + \varrho(\boldsymbol{\nu}, \boldsymbol{\mu})) \right. \\ & \left. + \mathbb{E}\_{\boldsymbol{u}^{+}} \, \mathbb{1} \Omega(\boldsymbol{u} + \varrho(\boldsymbol{\nu}, \boldsymbol{\mu})) \tilde{+} \mathcal{T}\_{\boldsymbol{u} + \varrho(\boldsymbol{\nu}, \boldsymbol{\mu})^{-}}^{\mathcal{G}} \, \mathbb{1} \Omega(\boldsymbol{u}) \right]. \end{split} \tag{28}$$

If Ψ is preconcave *F*·*I*·*V*·*F*, then inequality (28) is reversed.

**Proof.** Since Ψ is a preinvex *F*·*I*·*V*·*F*, then for *θ* ∈ [0, 1] , we have

$$\begin{split} \Psi\_{\*}\left(\frac{2u+\varrho(\nu,\mu)}{2},\theta\right) &\leq \frac{1}{2} \left(\Psi\_{\*}\left(u+\left(1-\xi\right)\varrho(\nu,\mu)\right),\theta\right) + \Psi\_{\*}\left(u+\xi\varrho(\nu,\mu)\right),\theta\right) \\ \Psi\_{\*}^{\*}\left(\frac{2u+\varrho(\nu,\mu)}{2},\theta\right) &\leq \frac{1}{2} \left(\Psi^{\*}\left(u+\left(1-\xi\right)\varrho(\nu,\mu)\right),\theta\right) + \Psi^{\*}\left(u+\xi\varrho(\nu,\mu),\theta\right) \end{split} \tag{29}$$

Since Ω(*u* + (1 − *ς*)*ϕ*(*ν*, *u*)) = Ω(*u* + *ςϕ*(*ν*, *u*)), then by multiplying (29) by *ς β*−1Ω(*u* + *ςϕ*(*ν*, *u*)) and integrate it with respect to *ς* over [0, 1] , we obtain

$$\begin{split} & \quad \Psi\_{\*}\left(\frac{2u+\varrho(\nu,\mu)}{2},\theta\right) \int\_{0}^{1} \xi^{\beta-1} \Omega(u+\xi\varrho(\nu,u))d\xi \\ & \leq \frac{1}{2} \Biggl( \begin{aligned} &\int\_{0}^{1} \xi^{\beta-1} \Psi\_{\*}\left(u+\left(1-\xi\right)\varrho(\nu,u)\right),\theta\rangle \Omega(u+\xi\varrho(\nu,u))d\xi \\ &+ \int\_{0}^{1} \xi^{\beta-1} \Psi\_{\*}\left(u+\xi\varrho(\nu,u)\right),\theta\rangle \Omega(u+\xi\varrho(\nu,u))d\xi \end{aligned} \right) . \tag{30} \\ & \quad \begin{aligned} &\Psi^{\*}\left(\frac{2u+\varrho(\nu,u)}{2},\theta\right) \int\_{0}^{1} \Omega(u+\xi\varrho(\nu,u))d\xi \\ &\leq \frac{1}{2} \Biggl( &\int\_{0}^{1} \xi^{\beta-1} \Psi^{\*}\left(u+\left(1-\xi\right)\varrho(\nu,u)\right),\theta\rangle \Omega(u+\xi\varrho(\nu,u))d\xi \\ &+ \int\_{0}^{1} \xi^{\beta-1} \Psi^{\*}\left(u+\xi\varrho(\nu,u)\right),\theta\rangle \Omega(u+\xi\varrho(\nu,u))d\xi \end{aligned} \right). \end{split} \tag{31}$$

Let *ω* = *u* + *ςϕ*(*ν*, *u*).  (31)

Then from (31), we have

Γ(*β*) (*ϕ*(*ν*,*u*))*β* Ψ∗ 2*u*+*ϕ*(*ν*,*u*) 2 , *θ* I *β u*+ Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ω(*u*) ≤ Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗Ω(*u*) Γ(*β*) (*ϕ*(*ν*,*u*))*β* Ψ∗ 2*u*+*ϕ*(*ν*,*u*) 2 , *θ* I *β u*+ Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ω(*u*) ≤ Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗Ω(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗Ω(*u*) ,

from which, we have

$$\begin{split} &\frac{\Gamma(\boldsymbol{\theta})}{\left(\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\nu})\right)^{\mathsf{F}}} \Big[\Psi\_{\ast}\Big{(}\frac{2\boldsymbol{\nu}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})}{2},\boldsymbol{\theta}\Big{)},\ \Psi^{\ast}\Big{(}\frac{2\boldsymbol{\nu}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})}{2},\boldsymbol{\theta}\Big{)}\Big{]} \Big{[}\mathcal{I}^{\mathsf{F}}\_{\boldsymbol{u}^{\mathsf{F}}}\,\Omega(\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu}))\ \ +\mathcal{I}^{\mathsf{F}}\_{\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})}{\left(\boldsymbol{\nu}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})\right)^{\mathsf{F}}}\,\Omega(\boldsymbol{u})\Big{]} \\ &\leq \inf\_{\boldsymbol{\Lambda}\left(\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})\right)^{\mathsf{F}}} \Big[\mathcal{I}^{\mathsf{F}}\_{\boldsymbol{u}^{\mathsf{F}}}\,\Psi\_{\ast}\Omega(\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu}))\ \ +\mathcal{I}^{\mathsf{F}}\_{\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})}\,\left.\Psi\_{\ast}\Omega(\boldsymbol{u})\right\}\,\mathcal{I}^{\mathsf{F}}\_{\boldsymbol{u}^{\mathsf{F}}}\,\Psi^{\ast}\Omega(\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu}))\ +\mathcal{I}^{\mathsf{F}}\_{\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})}\,\left.\Psi^{\ast}\Omega(\boldsymbol{u})\right],\ \end{split}$$

that is

$$\begin{aligned} &\frac{\Gamma(\boldsymbol{\beta})}{\left(\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})\right)^{\boldsymbol{\beta}}}\Psi\left(\frac{2\boldsymbol{\mu}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})}{2}\right)\left[\mathcal{T}\_{\boldsymbol{\mu}^{+}}^{\boldsymbol{\beta}}\,\boldsymbol{\Omega}(\boldsymbol{\nu}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu}))\right.+\mathcal{T}\_{\boldsymbol{\mu}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})^{-}}^{\boldsymbol{\beta}}\,\boldsymbol{\Omega}(\boldsymbol{\mu})\right] \\ &\prec \frac{\Gamma(\boldsymbol{\beta})}{\left(\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})\right)^{\boldsymbol{\beta}}}\left[\mathcal{T}\_{\boldsymbol{\mu}^{+}}^{\boldsymbol{\beta}}\,\boldsymbol{\Psi}\boldsymbol{\Omega}(\boldsymbol{\nu}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu}))\right.+\mathcal{T}\_{\boldsymbol{\mu}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{\mu})^{-}}^{\boldsymbol{\beta}}\,\boldsymbol{\Psi}\boldsymbol{\Omega}(\boldsymbol{\nu})\right] \end{aligned}$$

This completes the proof.

**Example 3.6.** We consider the *F*·*I*·*V*·*F* Ψ : [0, 2] → F0 defined by,

$$\Psi(\omega)(\sigma) = \begin{cases} \frac{\sigma}{2 - \sqrt{\omega}\prime} & \sigma \in \left[0, 2 - \sqrt{\omega}\right] \\\\ \frac{2\left(2 - \sqrt{\omega}\right) - \sigma}{2 - \sqrt{\omega}} & \sigma \in \left(2 - \sqrt{\omega}, 2\left(2 - \sqrt{\omega}\right)\right] \\\\ 0, & \text{otherwise}. \end{cases}$$

Then, for each *θ* ∈ [0, 1] , we have Ψ*θ* (*ω*) = - *θ* 2 − √ *ω* , (2 − *θ*) 2 − √ *ω* . Since end point functions Ψ∗(*ω*, *θ*) , Ψ∗ (*ω*, *θ*) are preinvex functions with respect to *ϕ*(*ν*, *u*) = *ν* − *u* for each *θ* ∈ [0, 1], then Ψ(*ω*) is preinvex *F*·*I*·*V*·*F*. If

$$\Omega(\omega) := \left\{ \begin{array}{c} \sqrt{\omega} \\ \sqrt{2-\omega} \end{array} \begin{array}{c} \sigma \in \left[0,1\right] \\ \sigma \in \left(1,2\right] \end{array} \right\}$$

then Ω(2 − *ω*) = Ω(*ω*) ≥ 0, for all *ω* ∈ [0, 2]. Since Ψ∗(*ω*, *θ*) = *θ* 2 − √ *ω* and Ψ∗ (*ω*, *θ*) = (2 − *θ*) 2 − √ *ω* . If *β* = 1 2 , then we compute the following:

$$
\begin{split}
\left[T^{\theta}\_{\mu^{+}}\,\Psi\Omega(u+\varrho(\nu,u))\right]\hat{+}T^{\theta}\_{\mu+\varrho(\nu,u)}\cdots\Psi\Omega(u)\bigg] &\leqslant \frac{\mathbf{v}\_{\mu}(u)\cdot\mathbf{\varpi}(u+\varrho(\nu,u))}{2}\bigg[-\mathcal{R}^{\theta}\_{\mu+\varrho(\nu,u)}\cdot\Omega(u)\bigg] \\
\leqslant \frac{\mathbf{v}\_{\mu}(u)\cdot\mathbf{\varpi}(u)}{2}\bigg[\mathcal{T}^{\theta}\_{\mu^{+}}\,\Omega(u+\varrho(\nu,u))\bigg] + \mathcal{T}^{\theta}\_{\mu+\varrho(\nu,u)}\cdot\Omega(u)\bigg] \\
\dfrac{\mathbf{v}\_{\mu}(u)\cdot\mathbf{\varpi}\_{\begin{subarray}{c}\mu+\Psi(\nu,u)\\\Delta\end{subarray}}\bigg[T^{\theta}\_{\mu^{+}}\,\Omega(u+\varrho(\nu,u))\bigg] + \mathcal{T}^{\theta}\_{\mu+\varrho(\nu,u)}\cdot\Omega(u)\bigg]}{2} \\
\dfrac{\mathbf{v}\_{\mu}(u)\cdot\mathbf{\varpi}^{+}(u+\varrho(\nu,u))}{2}\bigg[T^{\theta}\_{\mu^{+}}\,\Omega(u+\varrho(\nu,u))\bigg] + \mathcal{T}^{\theta}\_{\mu+\varrho(\nu,u)}\cdot\Omega(u)\bigg] = \frac{\pi}{\sqrt{2}}\theta\left(\frac{4-\sqrt{2}}{2}\right),
\end{split}
\tag{32}
$$

$$
\dfrac{\frac{\mathbf{v}\_{\mu}(u)\cdot\mathbf{\varpi}^{+}(u+\varrho(\nu,u))}{2}\bigg[T^{\theta}\_{\mu^{+}}\,\Omega(u+\varrho(\nu,u))\bigg] + \mathcal{T}^{\theta}\_{\mu+\varrho(\nu,u)}\cdot\Omega(u)\bigg] = \frac{\pi}{\sqrt{2}}(2-\theta)\left(\frac{4-\sqrt{2}}{2}\right),
$$

$$\begin{aligned} \left[ \begin{aligned} \mathcal{T}\_{\boldsymbol{u}^{+}}^{\boldsymbol{\theta}} \,\Psi\_{\*}\Omega(\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{u})) &+ \mathcal{T}\_{\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{u})^{-}}^{\boldsymbol{\theta}} \,\Psi\_{\*}\Omega(\boldsymbol{u}) \right] &= \frac{1}{\sqrt{\pi}} \theta\left(2\pi+\frac{4-8\sqrt{2}}{3}\right), \\ \left[ \begin{aligned} \mathcal{T}\_{\boldsymbol{u}^{+}}^{\boldsymbol{\theta}} \,\Psi^{\*}\Omega(\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{u})) &+ \mathcal{T}\_{\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{u})^{-}}^{\boldsymbol{\theta}} \,\Psi^{\*}\Omega(\boldsymbol{u}) \right] &= \frac{1}{\sqrt{\pi}} (2-\theta)\left(2\pi+\frac{4-8\sqrt{2}}{3}\right). \end{aligned} \end{aligned} \tag{34}$$

From (32), (33) and (34), we have

$$\begin{aligned} \frac{1}{\sqrt{\pi}} \Big[ \theta \Big( 2\pi + \frac{4-8\sqrt{2}}{3} \Big), \ (2-\theta) \Big( 2\pi + \frac{4-8\sqrt{2}}{3} \Big) \Big] & \quad \leq \ \,\_1\frac{\pi}{\sqrt{2}} \Big[ \theta \Big( \frac{4-\sqrt{2}}{2} \Big), \ (2-\theta) \Big( \frac{4-\sqrt{2}}{2} \Big) \Big] \\ &= \frac{\pi}{\sqrt{2}} \Big[ \theta \Big( \frac{4-\sqrt{2}}{2} \Big), \ (2-\theta) \Big( \frac{4-\sqrt{2}}{2} \Big) \Big] \end{aligned}$$

for each *θ* ∈ [0, 1] . Hence, Theorem 10 is verified.

For Theorem 11, we have

$$\begin{split} \Psi\_{\*}\left(\frac{2\mu+\varrho(\nu,\mu)}{2},\theta\right) \left[\mathcal{T}\_{\boldsymbol{u}^{+}}^{\boldsymbol{\beta}}\,\Omega(\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{u})) + \mathcal{T}\_{\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{u})^{-}}^{\boldsymbol{\beta}}\,\Omega(\boldsymbol{u})\right] &= \theta\sqrt{\pi}, \\ \Psi^{\*}\left(\frac{2\mu+\varrho(\boldsymbol{\nu},\boldsymbol{u})}{2},\theta\right) \left[\mathcal{T}\_{\boldsymbol{u}^{+}}^{\boldsymbol{\beta}}\,\Omega(\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{u})) + \mathcal{T}\_{\boldsymbol{u}+\boldsymbol{\varrho}(\boldsymbol{\nu},\boldsymbol{u})^{-}}^{\boldsymbol{\beta}}\,\Omega(\boldsymbol{u})\right] &= (2-\theta)\sqrt{\pi}. \end{split} \tag{35}$$

From (34) and (35), we have √ *π*[*θ*, (2 − *θ*)] ≤*I* √ 1 *π* [*θ*(2*π* + 4−8 √ 2 3 ),(2 − *θ*)(2*π* + 4−8 √ 2 3 )] , for each *θ* ∈ [0, 1] .

**Remark 3.7.** If Ω(*ω*) = 1. Then from Theorem 3.4 and Theorem 3.5, we get Theorem 3.1.

Let *β* = 1. Then we obtain following *H*·*H* Fejér type inequality for preinvex *F*·*I*·*V*·*F*, see [28].

$$\mathbb{P}\left(\frac{2u+\varrho(\nu,\mu)}{2}\right) \preccurlyeq \frac{1}{\int\_{u}^{u+\varrho(\nu,\mu)} \Omega(\omega)d\omega} \ (\mathsf{FR}) \int\_{u}^{u+\varrho(\nu,\mu)} \Psi(\omega)\Omega(\omega)d\omega \preccurlyeq \frac{\Psi(u)+\Psi(\nu)}{2}$$

If Ψ∗(*ω*, *θ*) = Ψ∗ (*ω*, *θ*) with *ϕ*(*ω*, *y*) = *ω* − *y* and Ω(*ω*) = *β* = 1 = *θ*. Then from Theorem 3.4 and Theorem 3.5, we get the classical *H*·*H* inequality.

If Ψ∗(*ω*, *θ*) = Ψ∗ (*ω*, *θ*) with *ϕ*(*ω*, *y*) = *ω* − *y* and *β* = 1, then from Theorem 3.4 and Theorem 3.5, we obtain the classical *H*·*H* Fejér inequality, see [46].

From Theorems 3.8 and 3.9, now we get several fuzzy-interval fractional integral inequalities linked to fuzzy-interval fractional *H*·*H* type inequality for the product of preinvex *F*·*I*·*V*·*Fs*.

**Theorem 3.8.** *Let* Ψ, Φ : [*u*, *u* + *ϕ*(*ν*, *u*)] → F0 *be two preinvex F*·*I*·*V*·*Fs on* [*u*, *u* + *ϕ*(*ν*, *u*)] , *whose θ-levels* Ψ*θ* , Φ*θ* : [*u*, *u* + *ϕ*(*ν*, *u*)] ⊂ R → K*C* + *are defined by* Ψ*θ* (*ω*) = [Ψ∗(*ω*, *θ*) , Ψ∗ (*ω*, *θ*)] *and* Φ*θ* (*ω*) = [Φ∗(*ω*, *θ*) , Φ∗ (*ω*, *θ*)] *for all ω* ∈ [*u*, *u* + *ϕ*(*ν*, *u*)] *and for all θ* ∈ [0, 1]*. If* Ψ×eΦ ∈ *L*([*u*, *u* + *ϕ*(*ν*, *u*)] , F0) *and ϕ satisfies Condition C, then*

$$\frac{\Gamma(\beta)}{2(\varrho(\boldsymbol{\nu},\boldsymbol{\mu}))^{\mathsf{F}}} \left[ \mathcal{T}\_{\boldsymbol{\mu}^{+}}^{\mathsf{F}} \, \Psi(\boldsymbol{\mu} + \varrho(\boldsymbol{\nu},\boldsymbol{\mu})) \widetilde{\boldsymbol{\chi}} \, \Phi(\boldsymbol{\nu} + \varrho(\boldsymbol{\nu},\boldsymbol{\mu})) \widetilde{+} \mathcal{T}\_{\boldsymbol{\mu} + \varrho(\boldsymbol{\nu},\boldsymbol{\mu})}^{\mathsf{F}} \Psi(\boldsymbol{\mu}) \widetilde{\boldsymbol{\chi}} \Phi(\boldsymbol{\mu}) \right] $$

$$\prec \left( \frac{1}{2} - \frac{\beta}{(\beta+1)(\beta+2)} \right) \Delta \left( \boldsymbol{\mu}, \boldsymbol{\mu} + \varrho(\boldsymbol{\nu}, \boldsymbol{\mu}) \right) \widetilde{+} \left( \frac{\beta}{(\beta+1)(\beta+2)} \right) \nabla \left( \boldsymbol{\mu}, \boldsymbol{\mu} + \varrho(\boldsymbol{\nu}, \boldsymbol{\mu}) \right)$$

where ∆ (*u*, *u* + *ϕ*(*ν*, *u*)) = Ψ(*u*)×eΦ(*u*) +e Ψ(*u* + *ϕ*(*ν*, *u*))×eΦ(*u* + *ϕ*(*ν*, *u*)) , ∇ (*u*, *u*+ *ϕ*(*ν*, *u*)) = Ψ(*u*)×eΦ(*u* + *ϕ*(*ν*, *u*)) +e Ψ(*u* + *ϕ*(*ν*, *u*))×eΦ(*u*) , and ∆*θ* (*u*, *u* + *ϕ*(*ν*, *u*)) = [∆∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) , ∆ ∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*)] and ∇*θ* (*u*, *ν*) = [∇∗((*u*, *u* + *ϕ*(*ν*, *u*)), *θ*), ∇∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*)].

**Proof.** Since Ψ, Φ both are preinvex *F*·*I*·*V*·*Fs* and Condition C holds *ϕ* for then, for each *θ* ∈ [0, 1] we have

$$\begin{aligned} \Psi\_\*(\boldsymbol{u} + (1 - \boldsymbol{\xi})\boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) &= \Psi\_\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) + \boldsymbol{\xi}\boldsymbol{\varrho}(\boldsymbol{u}, \boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u})) \; , \; \boldsymbol{\theta}) \\ &\leq \boldsymbol{\zeta}\Psi\_\*(\boldsymbol{u}, \boldsymbol{\theta}) + (1 - \boldsymbol{\zeta})\Psi\_\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \\ \Psi^\*(\boldsymbol{u} + (1 - \boldsymbol{\zeta})\boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) &= \Psi^\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) + \boldsymbol{\xi}\boldsymbol{\varrho}(\boldsymbol{u}, \boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u})) \; , \; \boldsymbol{\theta}) \\ &\leq \boldsymbol{\zeta}\Psi^\*(\boldsymbol{u}, \boldsymbol{\theta}) + (1 - \boldsymbol{\zeta})\Psi^\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \; . \end{aligned}$$

and

$$\begin{aligned} \Phi\_\*(\boldsymbol{u} + (1 - \boldsymbol{\xi})\boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) &= \Phi\_\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) + \boldsymbol{\zeta}\boldsymbol{\varrho}(\boldsymbol{u}, \boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u})) \; , \; \boldsymbol{\theta}) \\ &\leq \boldsymbol{\xi}\Phi\_\*(\boldsymbol{u}, \boldsymbol{\theta}) + \left(1 - \boldsymbol{\zeta}\right)\Phi\_\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \\\\ \Phi^\*(\boldsymbol{u} + (1 - \boldsymbol{\zeta})\boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) &= \Phi^\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) + \boldsymbol{\zeta}\boldsymbol{\varrho}(\boldsymbol{u}, \boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u})) \; , \; \boldsymbol{\theta}) \\\\ &\leq \boldsymbol{\xi}\Phi^\*(\boldsymbol{u}, \boldsymbol{\theta}) + \left(1 - \boldsymbol{\zeta}\right)\Phi^\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \; . \end{aligned}$$

From the definition of preinvex *F*·*I*·*V*·*Fs* it follows that e0 4 Ψ(*ω*) and e0 4 Φ(*ω*), so

Ψ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) ≤ (*ς*Ψ∗(*u*, *θ*) + (1 − *ς*)Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*))(*ς*Φ∗(*u*, *θ*) + (1 − *ς*)Φ∗(*u* + *ϕ*(*ν*, *u*) , *θ*)) = *ς* 2Ψ∗(*u*, *θ*) × Φ∗(*u*, *θ*) + (1 − *ς*) 2Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) +*ς*(1 − *ς*)Ψ∗(*u*, *θ*) × Φ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) + *ς*(1 − *ς*)Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u*, *θ*) Ψ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) ≤ (*ς*Ψ∗ (*u*, *θ*) + (1 − *ς*)Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*))(*ς*Φ∗ (*u*, *θ*) + (1 − *ς*)Φ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*)) = *ς* 2Ψ∗ (*u*, *θ*) × Φ∗ (*u*, *θ*) + (1 − *ς*) 2Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) +*ς*(1 − *ς*)Ψ∗ (*u*, *θ*) × Φ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) + *ς*(1 − *ς*)Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u*, *θ*) , (36)

Analogously, we have

$$\begin{aligned} \Psi\_\*(\boldsymbol{u} + \boldsymbol{\xi}\boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \Phi\_\*(\boldsymbol{u} + \boldsymbol{\xi}\boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \\ \leq (1 - \boldsymbol{\zeta})^2 \Psi\_\*(\boldsymbol{u}, \boldsymbol{\theta}) \; \times \Phi\_\*(\boldsymbol{u}, \boldsymbol{\theta}) + \boldsymbol{\xi}^2 \Psi\_\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \; \times \Phi\_\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \\ + \boldsymbol{\xi}(1 - \boldsymbol{\zeta}) \Psi\_\*(\boldsymbol{u}, \boldsymbol{\theta}) \; \times \Phi\_\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) + \boldsymbol{\zeta}(1 - \boldsymbol{\zeta}) \Psi\_\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \; \times \Phi\_\*(\boldsymbol{u}, \boldsymbol{\theta}) \end{aligned} \tag{37}$$

$$\Psi^\*(\boldsymbol{u} + \boldsymbol{\xi}\boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta}) \; \times \Phi^\*(\boldsymbol{u} + \boldsymbol{\xi}\boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \; , \; \boldsymbol{\theta})$$

$$\leq (1 - \varsigma)^2 \mathbf{Y}^\*(\boldsymbol{u}, \boldsymbol{\theta}) \times \boldsymbol{\Phi}^\*(\boldsymbol{u}, \boldsymbol{\theta}) + \boldsymbol{\zeta}^2 \mathbf{Y}^\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \,, \boldsymbol{\theta}) \times \boldsymbol{\Phi}^\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \,, \boldsymbol{\theta})$$

$$+ \boldsymbol{\xi} (1 - \boldsymbol{\zeta}) \mathbf{Y}^\*(\boldsymbol{u}, \boldsymbol{\theta}) \times \boldsymbol{\Phi}^\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \,, \boldsymbol{\theta}) + \boldsymbol{\zeta} (1 - \boldsymbol{\zeta}) \mathbf{Y}^\*(\boldsymbol{u} + \boldsymbol{\varrho}(\boldsymbol{\nu}, \boldsymbol{u}) \,, \boldsymbol{\theta}) \times \boldsymbol{\Phi}^\*(\boldsymbol{u}, \boldsymbol{\theta}) \,.$$

$$\text{Adding (36) and (37), we have}$$

Ψ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) ≤ h *ς* 2 + (1 − *ς*) 2 i [Ψ∗(*u*, *θ*) × Φ∗(*u*, *θ*) + Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ϕ*(*ν*, *u*) , *θ*)] +2*ς*(1 − *ς*)[Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u*, *θ*) + Ψ∗(*u*, *θ*) × Φ∗(*u* + *ϕ*(*ν*, *u*) , *θ*)] Ψ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) ≤ h *ς* 2 + (1 − *ς*) 2 i [Ψ∗ (*u*, *θ*) × Φ∗ (*u*, *θ*) + Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*)] +2*ς*(1 − *ς*)[Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u*, *θ*) + Ψ∗ (*u*, *θ*) × Φ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*)] . (38)

Taking multiplication of (38) by *ς β*−1 and integrating the obtained result with respect to *ς* over (0,1), we have

R 1 0 *ς β*−1Ψ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +*ς β*−1Ψ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*)*dς* ≤ ∆∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) R 1 0 *ς β*−1 h *ς* 2 + (1 − *ς*) 2 i *dς* +2∇∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) R 1 0 *ς β*−1 *ς*(1 − *ς*)*dς* R 1 0 *ς β*−1Ψ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +*ς β*−1Ψ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*)*dς* ≤ ∆ ∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) R 1 0 *ς β*−1 h *ς* 2 + (1 − *ς*) 2 i *dς* +2∇∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) R 1 0 *ς β*−1 *ς*(1 − *ς*)*dς*.

It follows that,

Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗(*u*, *θ*) × Φ∗(*u*, *θ*) ≤ 2 *β* 1 2 − *β* (*β*+1)(*β*+2) ∆∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) + 2 *β β* (*β*+1)(*β*+2) ∇∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) Γ(*β*) (*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗ (*u*, *θ*) × Φ∗ (*u*, *θ*) ≤ 2 *β* 1 2 − *β* (*β*+1)(*β*+2) ∆ ∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) + 2 *β β* (*β*+1)(*β*+2) ∇∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) , that is

Γ(*β*) (*ϕ*(*ν*,*u*))*β* [I *β u*+ Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ϕ*(*ν*, *u*) , *θ*) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗(*u*, *θ*) × Φ∗(*u*, *θ*) , I *β u*+ Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗ (*u*, *θ*) × Φ∗ (*u*, *θ*)] ≤*I* 2 *β* 1 2 − *β* (*β*+1)(*β*+2) [∆∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) , ∆ ∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*)] + 2 *β β* (*β*+1)(*β*+2) [∇∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) , ∇∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*)] Thus,

$$\frac{\Gamma(\beta)}{2(\varrho(\boldsymbol{\nu},\boldsymbol{\mu}))^{\widetilde{\beta}}} \left[ \mathcal{Z}\_{\boldsymbol{\mu}^{+}}^{\widetilde{\boldsymbol{\beta}}} \,\Psi(\boldsymbol{\nu}+\varrho(\boldsymbol{\nu},\boldsymbol{\mu})) \widetilde{\boldsymbol{\chi}} \,\Phi(\boldsymbol{\nu}+\varrho(\boldsymbol{\nu},\boldsymbol{\mu})) \widetilde{+} \mathcal{Z}\_{\boldsymbol{\mu}+\varrho(\boldsymbol{\nu},\boldsymbol{\mu})^{-}}^{\widetilde{\boldsymbol{\beta}}} \Psi(\boldsymbol{\nu}) \widetilde{\boldsymbol{\chi}} \Phi(\boldsymbol{\mu}) \right] $$

$$ \preceq \left( \frac{1}{2} - \frac{\beta}{(\beta+1)(\beta+2)} \right) \Delta \left( \boldsymbol{\mu}, \boldsymbol{\mu} + \varrho(\boldsymbol{\nu}, \boldsymbol{\mu}) \right) \widetilde{+} \left( \frac{\beta}{(\beta+1)(\beta+2)} \right) \nabla \left( \boldsymbol{\mu}, \boldsymbol{\mu} + \varrho(\boldsymbol{\nu}, \boldsymbol{\mu}) \right).$$

and the theorem has been established.

**Theorem 3.9.** *Let* Ψ, Φ : [*u*, *u* + *ϕ*(*ν*, *u*)] → F0 *be two preinvex F*·*I*·*V*·*Fs, whose θ-levels define the family of I*·*V*·*Fs* Ψ*θ* , Φ*θ* : [*u*, *u* + *ϕ*(*ν*, *u*)] ⊂ R → K*C* + *are given by* Ψ*θ* (*ω*) = [Ψ∗(*ω*, *θ*) , Ψ∗ (*ω*, *θ*)] *and* Φ*θ* (*ω*) = [Φ∗(*ω*, *θ*) , Φ∗ (*ω*, *θ*)] *for all ω* ∈ [*u*, *u* + *ϕ*(*ν*, *u*)] *and for all θ* ∈ [0, 1]*. If* Ψ×eΦ ∈ *L*([*u*, *u* + *ϕ*(*ν*, *u*)] , F0) *and ϕ satisfies Condition C, then*

$$\frac{1}{\beta} \le \left(\frac{2u + \varrho(\nu, u)}{2}\right) \widetilde{\times} \Phi\left(\frac{2u + \varrho(\nu, u)}{2}\right)$$

$$\prec \frac{\Gamma(\beta + 1)}{4(\varrho(\nu, u))^{\beta}} \left[\mathcal{Z}\_{u^{+}}^{\beta} \,\,\Psi(u + \varrho(\nu, u)) \,\widetilde{\times} \Phi(u + \varrho(\nu, u)) \,\widetilde{\div} \mathcal{I}\_{u + \varrho(\nu, u)}^{\beta} \cdot \Psi(u) \,\widetilde{\times} \Phi(u)\right]$$

$$\widetilde{+} \frac{1}{2\beta} \left(\frac{1}{2} - \frac{\beta}{(\beta + 1)(\beta + 2)}\right) \,\nabla \,(u, u + \varrho(\nu, u)) \widetilde{\div} \frac{1}{2\beta} \left(\frac{\beta}{(\beta + 1)(\beta + 2)}\right) \,\Delta\left(u, u + \varrho(\nu, u)\right)\,,$$

$$\text{where } \Delta\left(u, u + \varrho(\nu, u)\right) = \Psi(u) \widetilde{\times} \Phi(u) \,\,\widetilde{\times} \Psi(u + \varrho(\nu, u)) \,\widetilde{\times} \Phi(u + \varrho(\nu, u))\,,\,\nabla\left(u, u\right) = \Psi(u) \widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\widetilde{\times} \Psi(u) \,\,\$$

where ∆ (*u*, *u* + *ϕ*(*ν*, *u*)) = Ψ(*u*)×eΦ(*u*) +e Ψ(*u* + *ϕ*(*ν*, *u*))×eΦ(*u* + *ϕ*(*ν*, *u*)) , ∇ (*u*, *ν*) = Ψ(*u*)×eΦ(*u* + *ϕ*(*ν*, *u*)) +e Ψ(*u* + *ϕ*(*ν*, *u*))×eΦ(*u*) , and ∆*θ* (*u*, *u* + *ϕ*(*ν*, *u*)) = [∆∗((*u*, *u* + *ϕ*(*u* + *ϕ*(*ν*, *u*))) , *θ*) , ∆ ∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*)] and ∇*θ* (*u*, *u* + *ϕ*(*ν*, *u*)) = [∇∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) , ∇∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*)] .

**Proof.** Consider Ψ, Φ : [*u*, *u* + *ϕ*(*ν*, *u*)] → F0 are preinvex *F*·*I*·*V*·*Fs*. Then by hypothesis, for each *θ* ∈ [0, 1] , we have

Ψ∗ 2*u*+*ϕ*(*ν*,*u*) 2 , *θ* × Φ∗ 2*u*+*ϕ*(*ν*,*u*) 2 , *θ* Ψ∗ 2*u*+*ϕ*(*ν*,*u*) 2 , *θ* × Φ∗ 2*u*+*ϕ*(*ν*,*u*) 2 , *θ* ≤ 1 4 " Ψ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) # +1 4 " Ψ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) # ≤ 1 4 " Ψ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) # +1 4 Ψ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) , ≤ 1 4 " Ψ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) # +1 4 (*ς*Ψ∗(*u*, *θ*) + (1 − *ς*)Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*)) × ((1 − *ς*)Φ∗(*u*, *θ*) + *ς*Φ∗(*u* + *ϕ*(*ν*, *u*) , *θ*)) + ((1 − *ς*)Ψ∗(*u*, *θ*) + *ς*Ψ∗(*u* + *ϕ*(*ν*, *u*) , *θ*)) × (*ς*Φ∗(*u*, *θ*) + (1 − *ς*)Φ∗(*u* + *ϕ*(*ν*, *u*) , *θ*)) ≤ 1 4 " Ψ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) # +1 4 (*ς*Ψ∗ (*u*, *θ*) + (1 − *ς*)Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*)) × ((1 − *ς*)Φ∗ (*u*, *θ*) + *ς*Φ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*)) + ((1 − *ς*)Ψ∗ (*u*, *θ*) + *ς*Ψ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*)) × (*ς*Φ∗ (*u*, *θ*) + (1 − *ς*)Φ∗ (*u* + *ϕ*(*ν*, *u*) , *θ*)) , = 1 4 Ψ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗(*u* + *ςϕ*(*ν*, *u*) , *θ*) +1 4 n *ς* 2 + (1 − *ς*) 2 o ∇∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) + {*ς*(1 − *ς*) + (1 − *ς*)*ς*}∆∗((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) = 1 4 Ψ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + (1 − *ς*)*ϕ*(*ν*, *u*) , *θ*) +Ψ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) × Φ∗ (*u* + *ςϕ*(*ν*, *u*) , *θ*) +1 4 n *ς* 2 + (1 − *ς*) 2 o ∇∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) + {*ς*(1 − *ς*) + (1 − *ς*)*ς*}∆ ∗ ((*u*, *u* + *ϕ*(*ν*, *u*)) , *θ*) .

**Example 3.10.** Let [*u*, *u* + *ϕ*(*ν*, *u*)] = [0, *ϕ*(2, 0)], *β* = 1 2 , Ψ(*ω*) = [*ω*, 2*ω*], and Φ(*ω*) = [*ω*, 3*ω*].

$$\Psi(\omega)(\theta) = \begin{cases} \begin{array}{cc} \frac{\theta}{\omega} & \theta \in [0, \omega] \\ \frac{2\omega - \theta}{\omega} & \theta \in (\omega, 2\omega] \\ 0 & \text{otherwise} \end{array} \\\\ \Phi(\omega)(\theta) = \begin{cases} \begin{array}{cc} \frac{\theta}{2\omega} & \theta \in [0, 2\omega] \\ \frac{4\omega - \theta}{2\omega} & \theta \in (2\omega, 4\omega] \\ 0 & \text{otherwise} \end{array} \end{cases} \end{cases}$$

Then, for each *θ* ∈ [0, 1] , we have Ψ*θ* (*ω*) = [*θω*, (2 − *θ*)*ω*] and Φ*θ* (*ω*) = [2*θω*, 2(2 − *θ*)*ω*] . Since left and right end point functions Ψ∗(*ω*, *θ*) = *θω*, Ψ∗ (*ω*, *θ*) = (2 − *θ*)*ω*, Φ∗(*ω*, *θ*) = 2*θω* and Φ∗ (*ω*, *θ*) = 2(2 − *θ*)*ω* are preinvex functions with respect to *ϕ*(*ν*, *u*) = *ν* − *u* and for each *θ* ∈ [0, 1], then Ψ(*ω*) and Φ(*ω*) both are preinvex *F*·*I*·*V*·*F*. We clearly see that Ψ(*ω*)×eΦ(*ω*) ∈ *L*([*u*, *u* + *ϕ*(*ν*, *u*)] , F0) and

Γ(1+*β*) 2(*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗(*u* + *ϕ*(*ν*, *u*)) × Φ∗(*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗(*u*) × Φ∗(*u*) = Γ( 3 2 ) 2 √ 2 √ 1 *π ϕ*(2,0) R 0 (2 − *ω*) −1 2 2*θ* 2*ω*2 *dω* + Γ( 3 2 ) 2 √ 2 √ 1 *π ϕ*(2,0) R 0 (*ω*) −1 2 2*θ* 2*ω*2 *dω* ≈ 2.9332*θ* 2 , Γ(1+*β*) 2(*ϕ*(*ν*,*u*))*β* I *β u*+ Ψ∗ (*u* + *ϕ*(*ν*, *u*)) × Φ∗ (*u* + *ϕ*(*ν*, *u*)) + I *β u*+*ϕ*(*ν*,*u*) − Ψ∗ (*u*) × Φ∗ (*u*) = Γ( 3 2 ) 2 √ 2 √ 1 *π ϕ*(2,0) R 0 (2 − *ω*) −1 2 .2(2 − *θ*) 2*ω*2*dω* + Γ( 3 2 ) 2 √ 2 √ 1 *π ϕ*(2,0) R 0 (*ω*) −1 2 .2(2 − *θ*) 2*ω*2*dω* ≈ 2.9332(2 − *θ*) 2 ,

Note that

$$
\left(\frac{1}{2} - \frac{\beta}{(\beta+1)(\beta+2)}\right) \Delta\_\*(u, u + \varrho(\nu, u)) \\
= \left[\Psi\_\*(u) \times \Phi\_\*(u) + \Psi\_\*(u + \varrho(\nu, u)) \times \Phi\_\*(u + \varrho(\nu, u))\right]
$$

$$
= \frac{11}{30} \mathcal{S} \theta^2,
$$

$$
\left(\frac{1}{2} - \frac{\beta}{(\beta+1)(\beta+2)}\right) \Delta^\*(u, u + \varrho(\nu, u))
$$

$$
= \left[\Psi^\*(u) \times \Phi^\*(u) + \Psi^\*(u + \varrho(\nu, u)) \times \Phi^\*(u + \varrho(\nu, u))\right] \\
= \frac{11}{30} \mathcal{S} (2 - \theta)^2,
$$

$$
\left(\frac{\beta}{(\beta+1)(\beta+2)}\right) \nabla\_\*(u, u + \varrho(\nu, u)) \\
= \left[\Psi\_\*(u) \times \Phi\_\*(u + \varrho(\nu, u)) + \Psi\_\*(u + \varrho(\nu, u)) \times \Phi\_\*(u)\right]
$$

$$
= \frac{2}{15} \left(0\right),
\\ \left(\frac{\beta}{(\beta+1)(\beta+2)}\right) \nabla\_\*(u, u + \varrho(\nu, u)) \\
= \left[\Psi^\*(u) \times \Phi^\*(u + \varrho(\nu, u))\right] \\
\times \left[\Psi^\*(u) \frac{\beta}{(\beta+1)(\beta+2)}\right] \\
= \frac{2}{15} \left(0\right).
$$

Therefore, we have

$$
\left(\frac{1}{2} - \frac{\not\theta}{(\not\theta + 1)(\not\theta + 2)}\right) \Delta\_{\theta}(\left(\mu, \mu + \varrho(\nu, u)\right), \; \theta) \\
+ \left(\frac{\not\theta}{(\not\theta + 1)(\not\theta + 2)}\right) \nabla\_{\theta}(\left(\mu, \mu + \varrho(\nu, u)\right), \; \theta)
$$

$$
= \frac{11}{30} \left[8\theta^2, 8(2 - \theta)^2\right] \\
+ \frac{2}{15} [0, 0] \\
\approx \left[2.9332\theta^2, 2.9332(2 - \theta)^2\right].
$$

$$
\text{It follows that}
$$

$$\left[\left[2.9332\theta^2, 2.9332(2-\theta)^2\right] \le\_I \left[2.9332\theta^2, 2.9332(2-\theta)^2\right]\right] \text{ .}$$

and Theorem 3.7. has been demonstrated.

#### **4. Conclusions and Future Plan**

In this article, we established relation between integral inequalities and preinvex *F*·*I*·*V*·*Fs* using fuzzy Riemann–Liouville fractional integrals and Condition C. We addressed *H*·*H* type inequalities and *H*·*H* Fejér type inequalities for introduced preinvex *F*·*I*·*V*·*F.* Moreover, some related fuzzy fractional inequalities were also obtained. We gave useful examples to verify the validity of presented results. In future, we will try to explore this concept for generalized preinvex *F*·*I*·*V*·*Fs* and using fuzzy Riemann–Liouville fractional integrals, we will try to get new inequalities for preinvex *F*·*I*·*V*·*Fs.* We believe that the implications and methodologies presented in this article will energize and encourage scholars to pursue a more intriguing follow-up in this field. Finally, we think that our findings may be applied to other fractional calculus models having Mittag-Liffler functions

in their kernels, such as Atangana-Baleanue and Prabhakar fractional operators. This consideration has been kept as an open problem for academics interested in this topic. Researchers that are interested might follow the steps outlined in references [52,53].

**Author Contributions:** Conceptualization, M.B.K.; methodology, M.B.K.; validation, S.T., M.S.S. and H.G.Z.; formal analysis, J.E.M.-D.; investigation, M.S.S.; resources, S.T.; data curation, H.G.Z.; writing—original draft preparation, M.B.K., J.E.M.-D. and H.G.Z.; writing—review and editing, M.B.K. and S.T.; visualization, H.G.Z.; supervision, M.B.K. and M.S.S.; project administration, M.B.K.; funding acquisition, J.E.M.-D., M.S.S. and H.G.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research. This work was funded by Taif University Researchers Supporting Project number (TURSP-2020/345), Taif University, Taif, Saudi Arabia and this work was also supported by Consejo Nacional de Ciencia y Tecnología, Grant No. A1-S-45928.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Multiobjective Convex Optimization in Real Banach Space**

**Kin Keung Lai 1,\* ,†, Mohd Hassan 2,†, Jitendra Kumar Maurya 3,†, Sanjeev Kumar Singh 2,† and Shashi Kant Mishra 2,†**

**Abstract:** In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary and sufficient Pareto optimality conditions for considered problems under some constraint qualifications. These results are motivated by the symmetric results obtained in the recent article by Cobos Sánchez et al. in 2021 on Pareto optimality for multiobjective optimization problems of continuous linear operators. The discussions in this paper are also related to second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones due to Mishra and Wang in 2005. Further, we establish Karush–Kuhn– Tucker optimality conditions using saddle point optimality conditions for the differentiable cases and present some examples to illustrate our results. The study in this article can also be seen and extended as symmetric results of necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds by Ruiz-Garzón et al. in 2019.

**Keywords:** multiobjective programming; nonlinear programming; convex optimization; saddle point

#### **1. Introduction**

Consider the general multiobjective optimization problem

$$(\text{MOP})\text{ min }f(\mathbf{x}) = (f\_1(\mathbf{x}), \dots, f\_p(\mathbf{x})), \text{ subject to } \operatorname{g}(\mathbf{x}) \le 0, \, h(\mathbf{x}) = 0,\tag{1}$$

where the functions *f* : *X* → R*p* , *g* : *X* → R*q* , and *h* : *X* → R*r* are real vector valued functions and *X* is real Banach space.

Multiobjective optimization problem (MOP) arises when two or more objective functions are simultaneously optimized over a feasible region. The multiobjective optimization has been considerably analyzed and studied by many researchers, see for instance [1–6]. Multiobjective optimization problems play a crucial role in various fields like economics, engineering, management sciences [2,7–11], and many more places in daily life.

To deal with the multiobjective optimization problems, we have to find Pareto optimal solutions. These solutions are non-dominated by one another. A solution is called nondominated or Pareto optimal if none of the objective functions can be improved in value without reducing one or more objective values. One of the best techniques to deal with multiobjective optimization problems is scalarization. Wendell and Lee [12] developed the scalarization technique to deal with multiobjective optimization problems. Wendell and Lee [12] generalized the results on efficient points for multiobjective optimization problems to nonlinear optimization problems. The multiobjective problem is converted into a single objective problem in the scalarization technique.

The saddle point optimality conditions are briefly explained in [13], Rooyen et al. [14] constructed a Langrangian function for the convex multiobjective problem and established a relationship between saddle point optimality conditions and Pareto optimal solutions.

**Citation:** Lai, K.K.; Hassan, M.; Maurya, J.K.; Singh, S.K.; Mishra, S.K. Multiobjective Convex Optimization in Real Banach Space. *Symmetry* **2021**, *13*, 2148. https://doi.org/10.3390/ sym13112148

Academic Editors: Octav Olteanu and Savin Treanta

Received: 8 October 2021 Accepted: 27 October 2021 Published: 10 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Cobos-Sànchez et al. [15] proposed Pareto optimality conditions for multiobjective optimization problems of continuous linear operators. Recently, Treanta [16] studied robust saddle point criterion in second order partial differential equations and partial differential inequations.

Rooyen et al. [14] discussed necessary and sufficient optimality conditions for (MOP) without any constraint qualification in Euclidean space. Recently, Antczak and Abdulaleem [17] studied optimality and duality results for E-differentiable functions. Barbu and Precupanu [18] studied the saddle point optimality conditions of convex optimization problem for real Banach space. Valyi [19] proposed the concept of approximate saddle point condition for convex multiobjective optimization problems. Further Rong and Wu [20] generalized the results of Valyi [19] with set valued maps.

Karush–Kuhn–Tucker (KKT) optimality conditions [21] play a pivotal role to solve scalar optimization problem as well as multiobjective optimization problems. Recently Lai et al. [3] discussed unconstrained multiobjective optimization problems. Further, Guu et al. [22] studied strong KKT type sufficient optimality conditions for semi-infinite programming problems.

Motivated by the work of Barbu and Precupanu [18], Rooyen et al. [14] and Wendell and Lee [12], we extend the results related to saddle point optimality conditions and Karush–Kuhn–Tucker optimality conditions from single objective function to multiobjective function with the help of Slater's constraint qualifications [13]. We also present some illustrative examples to support the theory.

The organization of this paper is as follows: In Section 2, we recall some preliminaries and basic results. In Section 3, results on saddle point and Karush–Kuhn–Tucker necessary optimality conditions for multiobjective optimization problems are extended. Further, we establish the relationship between the Pareto solution and the saddle point for the Lagrange function using Slater's constraint qualification. The last section is dedicated to conclusions and future remarks.

#### **2. Preliminaries**

In this section, we recall some notions and preliminary results which will be used in this paper. R denotes the set of real numbers. Let *X* be real Banach Space and *X* ∗ be its dual space. Let *x*, *y* ∈ R*n* , then following inequalities represent their meaning as follows:

$$\begin{aligned} &x \geq y \implies x\_{i} \geq y\_{i} \\ &x \geq y \implies x\_{i} \geq y\_{i\prime} \; \mathbf{x} \neq y\_{\prime} \\ &x > y \implies x\_{i} > y\_{i\prime} \\ &x = y \implies x\_{i} = y\_{i} .\end{aligned}$$

We denote the feasible region as

$$S = \{ \mathfrak{x} \in X \colon \mathfrak{g}(\mathfrak{x}) \le \mathbf{0}, \ h(\mathfrak{x}) = \mathbf{0} \}.$$

To deal with the multiobjective optimization problems (MOP), we require some basic definitions.

**Definition 1** (Ref. [2])**.** *A decision vector x*¯ ∈ *S is global Pareto optimal solution (global efficient solution) if there does not exist another decision vector x* ∈ *S such that*

$$f(\mathfrak{x}) \le f(\mathfrak{x}).$$

Consider the following scalarized multiobjective optimization problem (SMOP) corresponding to (*MOP*):

$$(SMOP) \quad \min \sum\_{i=1}^{p} f\_i(x)$$

subject to *f*(*x*) 5 *f*(*x*¯), *g*(*x*) 5 0, *h*(*x*) = 0, where *x*¯ is any feasible point of (*MOP*).

Now, we recall the result from [2], which relate the solution of (*MOP*) and (*SMOP*).

**Theorem 1** (Ref. [7])**.** *A feasible point x*¯ ∈ *S is a Pareto optimal solution of the (MOP) if and only if x is an optimal solution of the (SMOP).* ¯

**Definition 2** (Ref. [13])**.** *A subset of the linear space X is said to be convex if for every distinct pair x and y of subset, it contains λx* + (1 − *λ*)*y*, ∀ *λ* ∈ [0, 1]*.*

**Definition 3** (Ref. [13])**.** *A function f is said to be convex on X if the inequality*

$$f(\lambda x + (1 - \lambda)y) \le \lambda f(x) + (1 - \lambda)f(y)$$

*holds for all x*, *y* ∈ *X and for every λ* ∈ [0, 1].

**Definition 4** (Ref. [18])**.** *The function f* : *X* → R = [−∞, +∞] *is said to be proper convex if f*(*x*) > −∞ ∀ *x* ∈ *X and if f is not the constant function then* +∞ *(that is, f* 6≡ +∞*). If f is a convex function, Dom*(*f*) *denotes the effective domain of f* , *which is as follows:*

$$Dom(f) = \{ \mathfrak{x} \in X : f(\mathfrak{x}) < +\infty \}.$$

*If f is proper, then Dom*(*f*) *is finite. Conversely, if A is a nonempty convex subset of X and f is a finite and convex function on A, then one can obtain a proper convex function on X by setting f*(*x*) = +∞ *if x* ∈ *X* \ *A*.

**Definition 5** (Ref. [18])**.** *The function f* : *X* → R *is called lower semi continuous at x if* ¯

$$f(\mathfrak{x}) = \lim\_{\mathfrak{x} \to \mathfrak{x}} \inf f(\mathfrak{x}).$$

**Corollary 1** (Ref. [18])**.** *If A*1 *and A*2 *are two non-empty disjoint convex sets of* R*n , there exist a non zero element c* = (*c*1, · · · , *cn*) ∈ R*n* \ {0} *such that*

$$\sum\_{i=1}^{n} c\_i u\_i \le \sum\_{i=1}^{n} c\_i v\_{1\prime} \,\,\forall\,\,u = (u\_i) \in A\_{1\prime} \,\,\forall\,\,v = (v\_i) \in A\_2.$$

**Definition 6** (Ref. [18])**.** *Given the proper convex function f* : *X* →] − ∞, +∞], *the subdifferential of such a function is the mapping ∂ f* : *X* → *X* ∗ *defined by*

$$\partial f(\mathbf{x}) = \{ \mathbf{x}^\* \in \mathcal{X}^\* : f(u) - f(\mathbf{x}) \ge (u - \mathbf{x}, \mathbf{x}^\*), \forall \, u \in X \}, \mathbf{x}$$

*where X* ∗ *is dual of X and* (., .) *denote the canonical pairing between X and X* ∗ . *The element x* ∗ ∈ *∂ f*(*x*) *called subgradient of f at x*.

**Corollary 2** (Ref. [18])**.** *If f is a proper convex function on X, then the minimum (global) of f over X is attained at the point x*¯ ∈ *X if and only if* 0 ∈ *∂ f*(*x*¯).

**Theorem 2** (Ref. [18])**.** *If the functions f*1 *and f*2 *are finite at a point in which at least one is continuous then*

$$
\partial (f\_1 + f\_2)(\mathbf{x}) = \partial f\_1(\mathbf{x}) + \partial f\_2(\mathbf{x}) \,\forall \,\mathbf{x} \in X.
$$

**Definition 7.** *Slater's constraint qualification*

$$0 \in \{ (h\_1(\mathfrak{x}), h\_2(\mathfrak{x}), \dots, h\_r(\mathfrak{x})); \mathfrak{x} \in \mathfrak{X}\_0 \}.$$

#### **3. Saddle Point and Karush–Kuhn–Tucker Optimality Conditions**

In this section, we established saddle point and Karush–Kuhn–Tucker type optimality conditions for considered (MOP) in Banach space.

 If x*¯ *is a Pareto optimal solution of the (MOP).  Let *x*¯ be an Pareto optimal solution of the consistent problem (MOP). Then, from Theorem 1 *x*¯ is an optimal solution of the problem

$$\begin{aligned} \left(^{\text{\*}}(\text{SMOP})\right)\min & \sum\_{i=1}^{p} f\_i(\mathbf{x})\\ \text{subject to } f\_i(\mathbf{x}) & \le f\_i(\mathbf{x}) \; \forall \; i = 1, \dots, p, \\ & g\_j(\mathbf{x}) \le 0 \; (j = 1, \dots, q), \; h\_k(\mathbf{x}) = 0 \; (k = 1, \dots, r). \end{aligned}$$

Now, we consider the subset

$$B = \left\{ \sum\_{i=1}^{p} f\_i(\mathbf{x}) - \sum\_{i=1}^{p} f\_i(\mathbf{\tilde{x}}) + a\_{0\prime}^f f\_1(\mathbf{x}) - f\_1(\mathbf{\tilde{x}}) + a\_{1\prime}^f \cdot \dots \cdot f\_p(\mathbf{x}) - f\_p(\mathbf{\tilde{x}}) + a\_{p\prime}^f \right.$$

$$g\_1(\mathbf{x}) + a\_{1\prime}^g \cdot \dots \cdot g\_q(\mathbf{x}) + a\_q^g \cdot h\_1(\mathbf{x}) \cdot \dots \cdot h\_r(\mathbf{x}) \colon \mathbf{x} \in \mathbf{X}\_0, a\_j^f > 0 \; \forall \ i, \ a\_j^g > 0 \; \forall \ j \right\}. \tag{3}$$

It is easy to see that the set *B* does not contain origin as well as it is a non-void convex subset of R1+*p*+*q*+*r* .  Again taking *x* = *x*¯, *α f i* ↓ 0 (∀ *i*), *α g j* ↓ 0 for *j* 6= *l* and *α g l* ↑ ∞, we get

$$
\lambda\_0^f \ge \mathbf{0}, \ \lambda\_i^f \ge \mathbf{0} \text{ and } \lambda\_i^g \ge \mathbf{0}.
$$

Thus, relation (4) becomes

$$
\lambda\_0^f \left\{ \sum\_{i=1}^p f\_i(\mathbf{x}) - \sum\_{i=1}^p f\_i(\mathbf{x}) \right\} + \sum\_{i=1}^p \lambda\_i^f \{f\_i(\mathbf{x}) - f\_i(\mathbf{x})\} + \sum\_{j=1}^q \lambda\_j^g g\_j(\mathbf{x}) + \sum\_{k=1}^r \lambda\_k^h h\_k(\mathbf{x}) \ge 0,
$$

$$
\implies \sum\_{i=1}^p f\_i(\mathbf{x}) \left\{ \lambda\_0^f + \lambda\_i^f \right\} + \sum\_{j=1}^q \lambda\_j^g g\_j(\mathbf{x}) \left. + \sum\_{k=1}^r \lambda\_k^h h\_k(\mathbf{x}) \right| \ge \sum\_{i=1}^p f\_i(\mathbf{x}) \left\{ \lambda\_0^f + \lambda\_i^f \right\},
$$

$$
\implies \sum\_{i=1}^p \lambda\_i^f f\_i(\mathbf{x}) + \sum\_{j=1}^q \lambda\_j^g g\_j(\mathbf{x}) + \sum\_{k=1}^r \lambda\_k^h h\_k(\mathbf{x}) \ge \sum\_{i=1}^p \lambda\_i^f f\_i(\mathbf{x}),
\tag{5}
$$

where *λ f i* = *λ*ˆ *f* 0 + *λ*ˆ *f i* . Since *x*¯ is feasible, therefore

$$
\lambda\_j^g \mathfrak{g}\_j(\bar{\mathfrak{x}}) \le \mathbf{0}, \forall \ j. \tag{6}
$$

substituting *x* = *x*¯ in inequality (5), we get

$$\sum\_{j=1}^{q} \lambda\_j^{\mathcal{S}} \mathfrak{g}\_j(\mathfrak{x}) \ge 0. \tag{7}$$

Now, from (6) and (7) we have *λ g j gj*(*x*¯) = 0, ∀ *j* = 1, . . . , *q*, which completes the proof.

*min f*(*x*) = (*f*1(*x*), *f*2(*x*)), *subject to g*(*x*) 5 0,

**Example 1.** *Consider the problem*

$$\begin{aligned} & \text{where } f\_1(\mathbf{x}) = \begin{cases} \mathbf{x}\_{1\prime}^2 & \text{if } -3 \le \mathbf{x}\_1, \mathbf{x}\_2 \le 3 \\ +\infty, & \text{otherwise} \end{cases}, f\_2(\mathbf{x}) = \begin{cases} \mathbf{x}\_{2\prime}^2 & \text{if } -3 \le \mathbf{x}\_1, \mathbf{x}\_2 \le 3 \\ +\infty, & \text{otherwise} \end{cases}, \\\ & \text{and } g(\mathbf{x}) = \begin{cases} (\mathbf{x}\_1 - 1)^2 + (\mathbf{x}\_2 - 1)^2 - 1, & \text{if } -3 \le \mathbf{x}\_1, \mathbf{x}\_2 \le 3 \\ +\infty, & \text{otherwise} \end{cases}. \end{aligned}$$

*Therefore, feasible region S* = {(*x*1, *x*2) ∈ R2 : (*x*1 − 1) 2 + (*x*2 − 1) 2 5 1} *and common effective domain X*0 = T2 *i*=1 *Dom*(*fi*) ∩ *Dom*(*g*) = {(*x*1, *x*2) ∈ R*n* : −1 5 *x*1, *x*2 5 1}. *Since, x*¯ = (1, 0) *is a Pareto optimal solution, then for λ f* 1 = 0, *λ f* 2 > 0, *λ g* = 0*, the following inequality satisfies*

$$\begin{split} &\lambda\_1^f f\_1(\bar{\mathbf{x}}) + \lambda\_2^f f\_2(\bar{\mathbf{x}}) = \mathbf{0} \leq \lambda\_1^f \mathbf{x}\_1^2 + \lambda\_2^f \mathbf{x}\_2^2 + \lambda^g [(\mathbf{x}\_1 - 1)^2 + (\mathbf{x}\_2 - 1)^2 - 1] \\ &= \lambda\_1^f f\_1(\mathbf{x}) + \lambda\_2^f f\_2(\mathbf{x}) + \lambda^g g(\mathbf{x}), \forall \, \mathbf{x} \in \mathcal{X}\_0. \end{split}$$

*Hence, result is verified.*

*λ*

Thus, it is natural to call the function

$$L(\mathbf{x}, \boldsymbol{\lambda}^f, \boldsymbol{\lambda}^g, \boldsymbol{\lambda}^h) = \sum\_{i=1}^p \boldsymbol{\lambda}\_i^f f\_i(\mathbf{x}) + \sum\_{j=1}^q \boldsymbol{\lambda}\_j^g \boldsymbol{g}\_j(\mathbf{x}) + \sum\_{k=1}^r \boldsymbol{\lambda}\_k^h h\_k(\mathbf{x}), \tag{8}$$

$$\boldsymbol{\lambda}^f = \langle \boldsymbol{\lambda}\_i^f \rangle \in \mathbb{R}^p, \boldsymbol{\lambda}^g = \langle \boldsymbol{\lambda}\_j^g \rangle \in \mathbb{R}^q \text{ and } \boldsymbol{\lambda}^h = \langle \boldsymbol{\lambda}\_k^h \rangle \in \mathbb{R}^r.$$

**Remark 1.** *The necessary conditions (2) with x*¯ ∈ *S are equivalent to the fact that the point* (*x*¯, *λ f* , *λ g* , *λ h* ) *is a saddle point for the Lagrange function (8) on X*0 × R*p* × R*q* × R*r , with respect to minimization on X*0 *and maximization on* R*p* × R*q* × R*r* , *that is,*