Bollini et al. [9] describe how the convolution of two ultradistributions of exponential type (UET) can exist. They then define the product of two UETs in terms of the convolution of their Fourier transforms. They obtain expressions for the Fourier transform of Lorentz invariant UETs (generalizing Bochner's theorem). Kami ´nski and Mincheva-Kaminska [10] present results concerning the convolution of distributions such as the existence of the convolution of tempered distributions whose supports are polynomially compatible sets. Ortner and Wagner [11] consider the Fourier transform of *O*(*p*, *q*) invariant distributions. They present a condition under which two Lorentz invariant tempered distributions are convolvable and a formula for their convolution.

Zinoviev [12] considers Lorentz invariant tempered distributions on (**R**4)*k* supported on the product of closed future light cones. Soloviev [13] discusses the theory of Lorentz covariant distributions, ultradistributions and hyperfunctions.

Harish-Chandra [14] realized the fruitfulness of regarding the space of invariant distributions as a module for the algebra of polynomial differential operators. In this context Kolk and Varadarajan [15] consider Lorentz invariant distributions supported on the boundary of the cone representing the causal future of the origin.

Our work does not consider the complexities and partial results of the general theory of Lorentz invariant distributions, ultradistributions and other such spaces but restricts attention to Lorentz invariant Borel complex measures. There are two reasons for this. Firstly one can obtain complete, unencumbered and "elegant" results. Secondly, many the distributional objects of interest in QFT (such as correlation functions) can, through Wick's theorem, or else the operator product expansion, be represented in terms of Feynman propagators and the propagators are Lorentz invariant measures (or else *K* invariant matrix-valued measures whose trace is Lorentz invariant) [16].

#### **3. The Feynman Scalar Field Propagator as a Tempered Measure**

In this section, we give a well-defined definition of the Feynman scalar propagator of QFT in terms of tempered measures and distributions. The propagator is viewed as being a complex tempered distribution. It is constructed from the Fourier transform of the tempered measure Ω*m*.

Consider the Feynman scalar field propagator. It is written as ([6], p. 35)

$$
\triangle\_F(\mathbf{x}) = -(2\pi)^{-4} \int \frac{e^{-ip\cdot\mathbf{x}}}{p^2 - m^2 + i\epsilon} \, dp. \tag{15}
$$

This is to be understood with respect to the *i*−epsilon procedure described in Mandl and Shaw ([17], p. 57), and the dot product *p*.*x* is given by

$$p.x = \eta\_{a\beta} p^a x^\beta \dots$$

where *η* = diag(1, −1, −1, −1). Therefore *F*(*x*) is written as

$$
\triangle\_F(\mathbf{x}) = -(2\pi)^{-4} \int\_{\mathbf{R}^3} \int\_{\mathbb{C}\_F} \frac{e^{-ip\cdot\mathbf{x}}}{p^2 - m^2} \, dp^0 d\stackrel{\rightarrow}{\mathcal{P}},\tag{16}
$$

where *CF* is the standard Feynman propagator contour. Thus *F*(*x*) is written as

$$
\triangle\_F(\mathbf{x}) = -(2\pi)^{-4} \int\_{\mathbb{R}^3} I(\overrightarrow{\mathcal{P}}, \mathbf{x}) \, d\overrightarrow{\mathcal{P}}\,\tag{17}
$$

where

$$I(\stackrel{\rightarrow}{p}, \mathbf{x}) = \int\_{C\_F} \frac{e^{-ip\cdot\mathbf{x}}}{(p^0)^2 - \omega\_m(\stackrel{\rightarrow}{p})^2} \, dp^0 \, \tag{18}$$

and

$$
\omega\_m(\stackrel{\smile}{p}) = (|\stackrel{\smile}{p}|^2 + m^2)^{\frac{1}{2}}, m \ge 0. \tag{19}
$$

$$
\downarrow \quad \downarrow \quad \downarrow \quad \downarrow \quad \downarrow \quad \downarrow \quad \downarrow
$$

The contour integral Equation (18) exists for *ωm*( *p* ) = 0 and is given by

$$I(\stackrel{\rightarrow}{\mathcal{P}},\mathbf{x}) = -\frac{\pi i}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}})} \begin{cases} \begin{array}{c} e^{-i(\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}})\mathbf{x}^{0} - \stackrel{\rightarrow}{\mathcal{P}}\mathbf{x})} \\ e^{-i(-\omega\_{\mathfrak{m}}(\mathcal{P})\mathbf{x}^{0} - \stackrel{\rightarrow}{\mathcal{P}}\mathbf{x})} \end{array} & \text{if } \mathbf{x}^{0} > 0, \\\ e^{-i(-\omega\_{\mathfrak{m}}(\mathcal{P})\mathbf{x}^{0} - \stackrel{\leftarrow}{\mathcal{P}}\mathbf{x})} & \text{if } \mathbf{x}^{0} < 0. \end{cases} \tag{20}$$

To prove this consider the contour *C*1(*R*) given by

$$\mathcal{C}\_1(\mathbb{R}) = \{ \mathcal{R}^{it} : 0 \le t \le \pi \}.$$

We will show that

$$I\_1(\mathbb{R}) = \int\_{\mathbb{C}\_1(\mathbb{R})} \frac{e^{-ip\cdot x}}{(p^0)^2 - \omega\_m(\overrightarrow{\mathcal{P}})^2} \, dp^0 \to 0 \text{ as } \mathbb{R} \to \infty,$$

as long as *x*0 < 0. To this effect we note that

$$\begin{split} |I\_{1}(R)| &= \left| \int\_{t=0}^{\pi} \frac{e^{-iR\hat{e}^{it}x^{0}} + i\stackrel{\rightarrow}{\mathcal{P}}\_{n}\overline{x}}{(R\epsilon^{it})^{2} - \omega\_{m}(\overrightarrow{\mathcal{P}})^{2}} iR\epsilon^{it}\,dt \right| \\ &\leq \int\_{t=0}^{\pi} \left| \frac{e^{R\sin t\hat{x}^{0}}}{(R\epsilon^{it})^{2} - \omega\_{m}(\overrightarrow{\mathcal{P}})^{2}} \right| \,R\,dt \\ &\leq \int\_{t=0}^{\pi} \frac{1}{|R^{2} - \omega\_{m}(\overrightarrow{\mathcal{P}})^{2}|} R \,dt \\ &= \frac{\pi R}{|R^{2} - \omega\_{m}(\overrightarrow{\mathcal{P}})^{2}|}, \\ &\to 0 \text{ as } R \to \infty, \text{ if } x^{0} < 0. \end{split} \tag{21}$$

Therefore, for *x*0 < 0,

$$I(\stackrel{\rightarrow}{p}, \mathbf{x}) = 2\pi i \text{res}(p^0 \mapsto \frac{e^{-ip\cdot\mathbf{x}}}{(p^0)^2 - \omega\_{\text{m}}(\stackrel{\rightarrow}{p})^2} \cdot \omega\_{\text{m}}(p) \cdot \mathbf{x}$$

Now

$$\frac{e^{-ip\cdot x}}{(p^0)^2 - \omega\_m(\vec{\mathcal{P}})^2} = \frac{e^{-ip\cdot x}}{(p^0 - \omega\_m(\vec{\mathcal{P}}))(p^0 + \omega\_m(\vec{\mathcal{P}}))} \cdot \frac{e^{-ip\cdot x}}{$$

Thus

$$I(\stackrel{\rightarrow}{p}, \mathbf{x}) = -\pi i \frac{e^{-i(-\omega\_m(\stackrel{\rightarrow}{p})\mathbf{x}^0 - \stackrel{\rightarrow}{p}\mathbf{.} \stackrel{\rightarrow}{\mathbf{x}})}}{\omega\_m(\stackrel{\rightarrow}{p})}$$

Similarly, if *x*0 > 0, then

$$I(\stackrel{\rightarrow}{p}, \mathbf{x}) = -\pi i \frac{e^{-i(\omega\_m(\stackrel{\rightarrow}{p})x^0 - \stackrel{\rightarrow}{p}.\stackrel{\rightarrow}{\mathbf{x}})}}{\omega\_m(\stackrel{\rightarrow}{p})}.$$

Hence

$$\int\_{\mathbb{R}^3} |I(\stackrel{\rightarrow}{\mathcal{P}}, \mathbf{x})| \, d\stackrel{\rightarrow}{\mathcal{P}} = \pi \int\_{\mathbb{R}^3} \frac{1}{\omega\_m(\stackrel{\rightarrow}{\mathcal{P}})} \, d\stackrel{\rightarrow}{\mathcal{P}} = \infty,\tag{22}$$

.

and so the integral Equation (17) defining *F*(*x*) does not exist as a Lebesgue integral.

We would like to give a well-defined interpretation of the propagator *F*. *H*± *m* for *m* ≥ 0 are orbits of the action of the Lorentz group on Minkowski space (these orbits correspond to real mass orbits, there are also "imaginary mass" hyperboloid orbits *Him*). Ω± *m* are Lorentz invariant measures for Minkowski space supported on *H*± *m* ([18], p. 157). Ω± *m* is locally finite for *m* ≥ 0. Now, for any non-negative measurable function *ψ* : **R**4 → [0, ∞],

$$\int\_{\mathbb{R}^4} \psi(p) \, \Omega\_m^{\pm}(dp) = \int\_{\mathbb{R}^3} \psi(\pm \omega\_m(\stackrel{\rightarrow}{\mathcal{P}}), \stackrel{\rightarrow}{\mathcal{P}}) \frac{d\stackrel{\rightarrow}{p}}{\omega\_m(\stackrel{\rightarrow}{\mathcal{P}})}.\tag{23}$$

Here, and for the rest of the section, the symbol *ψ* stands for a test function in Minkowski space. It follows from Equations (17), (20) and (23) that one may write

$$\triangle\_F(\mathbf{x}) = \begin{cases} (2\pi)^{-4}\pi \text{ri} \int e^{-ip\cdot\mathbf{x}} \, \Omega\_m^+(dp), & \text{if } \mathbf{x}^0 > 0, \\\ (2\pi)^{-4}\pi \text{ri} \int e^{-ip\cdot\mathbf{x}} \, \Omega\_m^-(dp), & \text{if } \mathbf{x}^0 < 0. \end{cases} \tag{24}$$

Equations (16), (17) and (24) are all integral expressions equivalent to Equation (15) and none of them exist as Lebesgue integrals. However, formally, Equation (24) can be written as

*F*(*x*) = ⎧ ⎨ ⎩ *πi* ∨ Ω+ *m*(−*x*) if *x*0 > 0, *πi* ∨ Ω− *m*(−*x*) if *x*0 < 0, (25)

where ∨ denotes the inverse Fourier transform operator (and we use the "physics"convention for the definition of the Fourier transform). Since Ω+ *m* and Ω− *m* are tempered distributions their inverse Fourier transforms exist and are tempered distributions. Let S±(**R**4) ⊂ S(**R**4) be the space of test functions supported in *S*±, where

$$S^{+} = \{ \mathbf{x} \in \mathbb{R}^{4} : \mathbf{x}^{0} > 0 \}, S^{-} = \{ \mathbf{x} \in \mathbb{R}^{4} : \mathbf{x}^{0} < 0 \}. \tag{26}$$

Then

$$<\triangle\_F, \psi> = \pi i < \Omega\_{\mathfrak{m}'}^\mp \psi> = \pi i < \Omega\_{\mathfrak{m}'}^\mp \stackrel{\vee}{\Psi}>,\tag{27}$$

for *ψ* ∈ S±(**R**4), where < *ω*, *ψ* > denotes the evaluation of a distribution *ω* on its test function argument *ψ*. Therefore the momentum space scalar field propagator on (S±(**R**4))∧ is

$$
\stackrel\frown{\triangle}\_F = \pi i \Omega\_m^\mp. \tag{28}
$$

(S+(**R**4))∧ is the space of wave functions with only positive frequency components while (S−(**R**4))∧ is the space of wave functions with only negative frequency components.

 Equations (15) and (25) lead to the *ansatz*

$$\frac{1}{p^2 - m^2 + i\epsilon} \to -\pi i \Omega\_m^{\pm}(p). \tag{29}$$

#### **4. A Spectral Calculus for Lorentz Invariant Measures**

Suppose that *μ* is a Lorentz invariant Borel complex measure on Minkowski space. Then by the spectral theorem, it must have the form of Equation (13). If *σ*2 = *σ*3 = 0 then *μ* will be said to be *causal* or a type I measure. If *σ*1 = *σ*3 = 0 then *μ* will be said to be a type II measure and if *c* = 0 and *σ*1 = *σ*2 = 0 then *μ* will be said to be a type III measure. Thus any Lorentz invariant measure is a sum of a type I measure, a type II measure and a type III measure. In particular, any measure of the form

$$\mu(\Gamma) = \int\_{m=0}^{\infty} \sigma(m) \Omega\_m^+(\Gamma) \, dm,\tag{30}$$

where *σ* is locally integrable function and the integration is carried out with respect to the Lebesgue measure, is a causal Lorentz invariant Borel complex measure. If *σ* is polynomially bounded then *μ* is a tempered measure.

The spectral calculus that we will now explain is a way to compute the spectrum *σ* of a Lorentz invariant measure *μ* if we know that *μ* can be written in the form of Equation (30) and *σ* is continuous.

For *m* > 0 and > 0 let *S*(*m*, ) be the hyperbolic (hyper-)disc defined by

$$S(m,\epsilon) = \{ p \in \mathbb{R}^4 : p^2 = m^2, |\vec{p}| < \epsilon, p^0 > 0 \},\tag{31}$$

where - *p* = *π*(*p*) = *π*(*p*0, *p*1, *p*2, *p*3)=(*p*1, *p*2, *p*3). For *a*, *b* ∈ **R** with 0 < *a* < *b* let Γ(*a*, *b*, ) be the hyperbolic cylinder defined by

$$\Gamma(a,b,\mathfrak{e}) = \bigcup\_{m \in (a,b)} \mathcal{S}(m,\mathfrak{e}).\tag{32}$$

Now suppose that we have a measure in the form of Equation (30) where *σ* is continuous. Then we can write

$$\begin{split} \mu(\Gamma(a,b,\epsilon)) &= \int\_{m=0}^{\infty} \sigma(m) \Omega\_{\mathfrak{m}}(\Gamma(a,b,\epsilon)) \, dm \\ &= \int\_{m=0}^{\infty} \sigma(m) \int\_{\pi(\Gamma(a,b,\epsilon)\cap\Pi\_{m}^{+})} \frac{d\stackrel{\rightarrow}{p}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\vec{p}})} \, dm \\ &= \int\_{a}^{b} \sigma(m) \int\_{B\_{\epsilon}(\mathbb{C})} \frac{\stackrel{\rightarrow}{\vec{p}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\vec{p}})} \, dm \\ &\approx \frac{4}{3} \pi \epsilon^{3} \int\_{a}^{b} \frac{\sigma(m)}{m} \, dm. \end{split} \tag{33}$$

where

$$
\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{p}}) = (\stackrel{\rightarrow}{\mathcal{p}}^2 + m^2)^{\frac{1}{2}},\tag{34}
$$

and *B*( - 0 ) = { - *p* ∈ **R**3 : | - *p* | < }. The approximation ≈ in the last line comes about because *ωm* is not constant over *B*( - 0 ). Thus if we define

$$\mathfrak{g}\_a(b) = \mathfrak{g}(a,b) = \lim\_{\epsilon \to 0} \epsilon^{-3} \mu(\Gamma(a,b,\epsilon)),\tag{35}$$

then we can retreive *σ* using the formula

$$
\sigma(b) = \frac{3}{4\pi} b \mathbf{g}\_a'(b). \tag{36}
$$

Thus we have proved the following fundamental theorem of the spectral calculus of causal Lorentz invariant measures.

**Theorem 2.** *Suppose that μ is a causal Lorentz invariant measure with continuous spectrum σ. Then σ can be calculated from the formula*

$$
\sigma(b) = \frac{3}{4\pi} b \mathbf{g}'\_a(b),
\tag{37}
$$

*where, for a*, *b* ∈ **R**, 0 < *a* < *b*, *ga* : (*a*, ∞) → **R** *is given by Equation (35).*

To make the proof of this theorem rigorous we prove the following.

**Lemma 1.** *Let a*, *b* ∈ **R**, 0 < *a* < *b. Then*

$$\lim\_{\epsilon \to 0} \epsilon^{-3} \int\_{B\_{\epsilon}(0)} \frac{d\stackrel{\rightarrow}{p}}{\omega\_{m}(\stackrel{\rightarrow}{p})} = \frac{4\pi}{3} \frac{1}{m'} \tag{38}$$

*uniformly for m* ∈ [*a*, *b*]*.*

**Proof.** Define

$$I = I(m, \epsilon) = \int\_{\mathcal{B}\_{\epsilon}(0)} \frac{d\stackrel{\rightarrow}{p}}{\omega\_{m}(\stackrel{\rightarrow}{p})}.\tag{39}$$

Then

$$I = \int\_{r=0}^{c} \frac{4\pi r^2 \, dr}{(r^2 + m^2)^{\frac{1}{2}}}.\tag{40}$$

Now

$$l\_1 < I < l\_{2\nu}$$

where

$$I\_1 = \int\_{r=0}^{\epsilon} \frac{4\pi r^2 \, dr}{(\epsilon^2 + m^2)^{\frac{1}{2}}} = \frac{4\pi}{(\epsilon^2 + m^2)^{\frac{1}{2}}} \frac{1}{3} \epsilon^3,$$

$$I\_2 = \int\_{r=0}^{\epsilon} \frac{4\pi r^2 \, dr}{m} = \frac{4\pi}{m} \frac{1}{3} \epsilon^3.$$

Therefore

$$\frac{4\pi}{3(\epsilon^2 + m^2)^{\frac{1}{2}}} < \epsilon^{-3}I < \frac{4\pi}{3m}.$$

Thus

$$
\frac{4\pi}{3m} - \frac{4\pi}{3(\epsilon^2 + m^2)^{\frac{1}{2}}} > \frac{4\pi}{3m} - \epsilon^{-3}I > 0.
$$

Hence

$$\left| \epsilon^{-3} I - \frac{4\pi}{3m} \right| < \frac{4\pi}{3m} - \frac{4\pi}{3(\epsilon^2 + m^2)^{\frac{1}{2}}}.\tag{41}$$

We have

$$\begin{split} \frac{4\pi}{3m} - \frac{4\pi}{3(\epsilon^2 + m^2)^{\frac{1}{2}}} &= \quad \frac{4\pi}{3} \frac{(\epsilon^2 + m^2)^{\frac{1}{2}} - m}{m(\epsilon^2 + m^2)^{\frac{1}{2}}} \\ &= \quad \frac{4\pi}{3} \frac{\epsilon^2}{m(\epsilon^2 + m^2)^{\frac{1}{2}}((\epsilon^2 + m^2)^{\frac{1}{2}} + m)} \\ &< \quad \frac{4\pi}{3} \frac{\epsilon^2}{2m^3} \\ &\leq \quad \frac{4\pi}{3} \frac{\epsilon^2}{2a^3}, \text{ for all } m \in [a, b]. \end{split}$$

Therefore

$$\left| \epsilon^{-3} I - \frac{4\pi}{3m} \right| < \frac{2\pi \epsilon^2}{3a^3},\tag{42}$$

for all *m* ∈ [*a*, *b*]

This lemma justifies the step of taking the limit under the integral sign (indicated by the symbol ≈) in the proof of Theorem 2.

More generally, suppose that *μ* : B0(**R**4) → **C** is a causal Lorentz invariant Borel measure on Minkowski space with spectrum *σ*. Then, by the Lebesgue decomposition theorem there exist unique measures *σc*, *σs* : B0([0, ∞)) → **C** such that *σ* = *σc* + *σs* where *σc*, the continuous part of the spectrum of *μ*, is absolutely continuous with respect to Lebesque measure and *σs*, the singular part of the spectrum of *μ*, is singular with respect to *σc*.

It is straightforward to prove the following.

**Theorem 3.** *Suppose that a* , *b* ∈ **R** *are such that* 0 < *a* < *b , σc*|(*a* ,*b* ) *is continuous. Then for all a*, *b* ∈ **R** *with a* < *a* < *b* < *b , ga*(*b*) *defined by Equation (35) exists and is continuously differentiable. Furthermore σc*|(*a* ,*b* ) *can be computed using the formula*

$$
\sigma\_c(b) = \frac{3}{4\pi} b g\_a'(b),
\tag{43}
$$

*and*

$$
\sigma\_s(E) = 0, \forall \text{ Borel } E \subset (a', b'). \tag{44}
$$

*Conversely suppose that a* , *b* ∈ **R** *are such that* 0 < *a* < *b and for all a*, *b* ∈ **R** *with a* < *a* < *b* < *b , ga*(*b*) *defined by Equation (35) exists and is continuously differentiable. Then σc*|(*a* ,*b* ) *is continuous and can be retrieved using the formula of Equation (43).*

#### **5. Investigation of the Measure Defined by the Convolution Ω***m ∗* **Ω***m*

*5.1. Determination of Some Properties of* Ω*m* ∗ Ω*m*

Consider the measure defined by

$$
\mu(\Gamma) = (\Omega\_{\mathfrak{m}} \* \Omega\_{\mathfrak{m}})(\Gamma) = \int \chi\_{\Gamma}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}}(dq), \tag{45}
$$

where, for any set Γ, *χ*Γ denotes the characteristic function of Γ defined by

$$\chi\_{\Gamma}(p) = \begin{cases} \begin{array}{l} 1 \text{ if } p \in \Gamma \\ 0 \text{ otherwise.} \end{array} \tag{46}$$

*μ* exists as a Borel measure because as |*p*|, |*q*| → ∞ with *p*, *q* ∈ *H*+ *m* , (*p* + *q*)0 → ∞ and so *p* + *q* is eventually /∈Γ for any compact set Γ ⊂ **R**4. Now

$$\begin{split} \mu(\Lambda(\Gamma)) &= \int \chi\_{\Lambda(\Gamma)}(p+q) \, \Omega\_m(dp) \, \Omega\_m(dq) \\ &= \int \chi\_{\Gamma}(\Lambda^{-1}p + \Lambda^{-1}q) \, \Omega\_m(dp) \, \Omega\_m(dq) \\ &= \int \chi\_{\Gamma}(p+q) \, \Omega\_m(dp) \, \Omega\_m(dq) \\ &= \mu(\Gamma), \end{split} \tag{47}$$

for all Λ ∈ *O*(1, 3)+↑, Γ ∈ B0(**R**4). Thus *μ* is a Lorentz invariant measure.

We will now show that *μ* is concentrated in the set

$$\mathcal{C}\_{2m} = \{ p \in \mathbb{R}^4 : p^2 \ge 4m^2, p^0 > 0 \}, \tag{48}$$

and therefore, that *μ* is causal. Let *U* ⊂ **R**4 be open. Then

$$
\mu(\mathcal{U}) = \int\_{\mathbb{R}^3} \int\_{\mathbb{R}^3} \chi\_{\mathcal{U}}(\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{V}}) + \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}), \stackrel{\rightarrow}{\mathcal{p}} + \stackrel{\rightarrow}{\mathcal{q}}) \frac{\stackrel{\rightarrow}{d\stackrel{\rightarrow}{p}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{p}})} \frac{\stackrel{\rightarrow}{d\stackrel{\rightarrow}{q}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})}.\tag{49}
$$

Therefore, using continuity, it follows that

$$
\mu(\iota\mathcal{U}) > 0 \quad \Leftrightarrow \quad (\exists \stackrel{\rightarrow}{\bar{q}}\_1 \stackrel{\rightarrow}{\bar{q}}\_2 \in \mathbb{R}^3) \\
\left(\omega\_m(\stackrel{\rightarrow}{\bar{q}}\_1) + \omega\_m(\stackrel{\rightarrow}{\bar{q}}\_2), \stackrel{\rightarrow}{\bar{q}}\_1 + \stackrel{\rightarrow}{\bar{q}}\_2\right) \in \mathcal{U}.
$$

Suppose that *p* ∈ supp(*μ*) (the support of the measure *μ*) i.e., *p* is such that *μ*(*U*) > 0 for all open neighborhoods *U* of *p*. Let *U* be an open neighborhood of *p*. Then, as *μ*(*U*) > 0, there exists *q* ∈ *U*, - *q* 1, - *q* 2 ∈ **R**3 such that *q* = (*ωm*( - *q* 1) + *ωm*( - *q* 2), - *q* 1 + - *q* 2). Clearly *q*0 ≥ 2*m*. As this is true for all neighborhoods *U* of *p* it follows that *p*0 ≥ 2*m*. By Lorentz invariance we may assume without loss of generality that - *p* = 0. Therefore *p*2 ≥ 4*m*2. Thus supp(*μ*) ⊂ *C*2*m*.

For the converse, let *p* = (*ωm*( - *p* ), - *p* ), *q* = (*ωm*( - *p* ), −- *p* ) ∈ *H*+ *m* for - *p* ∈ **R**3. As - *p* ranges over **R**3, *p* + *q* = (2*ωm*( - *p* ), - 0 ) ranges over {(*m* , - 0 ) : *m* ≥ 2*m*}. It follows using Lorentz invariance that supp(*μ*) ⊃ *C*2*m*.

Therefore the support supp(*μ*) of *μ* is *C*2*m*. Therefore by the spectral theorem *μ* has a spectral representation of the form

$$\mu(\Gamma) = \int\_{m'=2m}^{\infty} \Omega\_{m'}(\Gamma) \,\sigma(dm'),\tag{50}$$

for some Borel measure *σ* : B0([2*m*, ∞)) → **C**.

*5.2. Computation of the Spectrum of* Ω*m* ∗ Ω*m Using the Spectral Calculus*

Let *a*, *b* ∈ **R** with 0 < *a* < *b*. Let

$$\text{gg}\_a(b,\epsilon) = \mu(\Gamma(a,b,\epsilon)) \text{ for } \epsilon > 0. \tag{51}$$

We would like to calculate

$$\mathfrak{g}\_{a}(b) = \lim\_{\epsilon \to 0} \epsilon^{-3} \mathfrak{g}\_{a}(b, \epsilon), \tag{52}$$

and then retreive the spectral function as

$$
\sigma(b) = \frac{3}{4\pi} b \mathbf{g}'(b). \tag{53}
$$

To this effect we calculate

*g*(*a*, *b*, ) = *μ*(Γ(*a*, *b*, )) = *χ*Γ(*a*,*b*,)(*p* + *q*) Ω*m*(*dp*) Ω*m*(*dq*) ≈ *χ* (*a*,*b*)×*B*( - 0 ) (*p* + *q*) Ω*m*(*dp*) Ω*m*(*dq*) = *χ*(*a*,*b*)(*ωm*( - *p* ) + *ωm*( - *q* ))*χ B*( - 0 ) ( - *p* + - *q* ) *d* - *p ωm*( - *p* ) *d* - *q ωm*( - *q* ) = *χ*(*a*,*b*)(*ωm*( - *p* ) + *ωm*( - *q* ))*χ B*( - 0 )− - *q* ( - *p* ) *d* - *p ωm*( - *p* ) *d* - *q ωm*( - *q* ) ≈ *χ*(*a*,*b*)(2*ωm*( - *q* )) 4 3*π*3 *ωm*( - *q* )2 *d* - *q* .

We will call this argument Argument 1. This argument is intuitively reasonable but it needs to be justified rigorously. It is proved in the proof of Theorem 4 of the next section.

Given that Argument 1 is valid we now proceed to compute the spectrum of *μ*. We have

$$\begin{aligned} a < 2\omega\_{\mathfrak{m}}(\overrightarrow{\vec{q}}\,) < b &\iff \left(\frac{a}{2}\right)^2 - m^2 < \overrightarrow{\vec{q}}^2 < \left(\frac{b}{2}\right)^2 - m^2\\ \Leftrightarrow \quad mZ(a) < |\overrightarrow{\vec{q}}\,| < mZ(b), \end{aligned}$$

where

$$Z(m') = (\frac{m'^2}{4m^2} - 1)^{\frac{1}{2}}, \text{ for } m' \ge 2m. \tag{54}$$

Thus

$$g(a,b,\epsilon) \approx \frac{16\pi^2}{3}\epsilon^3 \int\_{r=mZ(a)}^{mZ(b)} \frac{r^2}{m^2 + r^2} dr. \tag{55}$$

Hence

$$g\_a(b) = \frac{16\pi^2}{3} \int\_{r=mZ(a)}^{mZ(b)} \frac{r^2}{m^2 + r^2} \, dr. \tag{56}$$

Therefore *ga* is continuously differentiable and so Theorem 3 applies. Using the Leibniz integral rule

$$g\_a'(b) = \frac{16\pi^2}{3} \frac{m^2 Z^2(b)}{m^2 + m^2 Z^2(b)} m Z'(b) = \frac{16\pi^2}{3} \frac{m Z(b)}{b}. \tag{57}$$

Therefore we compute the spectrum *σ* of *μ* as

$$\sigma(b) = \begin{cases} 4\pi m Z(b) \text{ for } b \ge 2m\\ 0 \text{ otherwise.} \end{cases} \tag{58}$$

#### **6. Proof of the Validity of Argument 1**

The following theorem establishes that Argument 1 is justified.

**Theorem 4.** *Let g*(*a*, *b*, ) *be defined by g*(*a*, *b*, ) = *μ*(Γ(*a*, *b*, )) *for a*, *b* ∈ **R**, *a* < *b*, > 0*, where μ* = Ω*m* ∗ Ω*m. Then the following formal argument (Argument 1)*

$$\begin{split} g(a,b,\epsilon) &= \quad \mu(\Gamma(a,b,\epsilon)) \\ &= \quad \int \chi\_{\Gamma(a,b,\epsilon)}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}}(dq) \\ &\approx \quad \int \chi\_{(a,b)\times B\_{\mathfrak{e}}(0)}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}}(dq) \\ &= \quad \int \chi\_{(a,b)}(\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}}) + \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})) \chi\_{B\_{\mathfrak{e}}(0)}(\stackrel{\rightarrow}{\mathcal{P}} + \stackrel{\rightarrow}{\mathcal{q}}) \, \frac{d\stackrel{\rightarrow}{\mathcal{P}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}})} \frac{d\stackrel{\rightarrow}{\mathcal{q}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}})} \\ &= \quad \int \chi\_{(a,b)}(\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}}) + \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})) \chi\_{\mathbb{B}\_{\mathfrak{e}}(0)}(\stackrel{\rightarrow}{\mathcal{q}}) \, \frac{d\stackrel{\rightarrow}{\mathcal{P}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}})} \frac{d\stackrel{\rightarrow}{\mathcal{q}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})} \\ &\approx \quad \int \chi\_{(a,b)}(2\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})) \, \frac{\frac{4}{3}\pi\epsilon^{3}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})^{2}} \, d\stackrel{\rightarrow}{\mathcal{q}}, \end{split}$$

*is justified in the sense that*

$$\lim\_{\epsilon \to 0} \epsilon^{-3} g(a, b, \epsilon) = \frac{4}{3} \pi \int \chi\_{(a, b)}(2\omega\_m(\stackrel{\rightarrow}{q})) \frac{1}{\omega\_m(\stackrel{\rightarrow}{q})^2} d\stackrel{\rightarrow}{\bar{q}}.\tag{59}$$

**Proof.** There are 2 ≈ signs that we have to consider. The first is in line 3 and arises because we are approximating the hyperbolic cylinder of radius between *a* and *b* with an ordinary cylinder of radius . We will show that the error is of order greater than 3. Let Γ = Γ(*a*, *b*, ) be the aforementioned hyperbolic cylinder. Then

$$\Gamma = \bigcup\_{m' \in (a,b)} \mathcal{S}(m', \mathfrak{e}). \tag{60}$$

Let

$$\begin{array}{rcl} \Gamma' &=& \bigcup\_{m' \in (a,b)} \{m'\} \times B\_{\varepsilon}(\stackrel{\rightarrow}{0}) = (a,b) \times B\_{\varepsilon}(\stackrel{\rightarrow}{0}) \\ \Gamma'^{-} &=& \bigcup\_{m' \in (a,a^{+})} \{(m',\stackrel{\rightarrow}{p}) : \stackrel{\rightarrow}{p}^{2} > m'^{2} - a^{2} \} \\ & & \bigcup\_{m' \in (a,a^{+})} \left( \{m'\} \times B\_{\varepsilon}(\stackrel{\rightarrow}{0}) \right) = (a,a^{+}) \times B\_{\varepsilon}(\stackrel{\rightarrow}{0}) \\ \Gamma'^{+} &=& \bigcup\_{m' \in (b,b^{+})} \left\{ (m',\stackrel{\rightarrow}{p}) : \stackrel{\rightarrow}{p}^{2} > m'^{2} - b^{2} \right\} \\ & & \subset \bigcup\_{m' \in (b,b^{+})} \left( \{m'\} \times B\_{\varepsilon}(\stackrel{\rightarrow}{0}) \right) = (b,b^{+}) \times B\_{\varepsilon}(\stackrel{\rightarrow}{0}), \end{array}$$

in which

$$a^{+} = (a^{2} + \epsilon^{2})^{\frac{1}{2}}, \\ b^{+} = (b^{2} + \epsilon^{2})^{\frac{1}{2}}.\tag{61}$$

Then Γ differs from (Γ ∼ Γ −) ∪ Γ + on a set of measure zero,

It is straightforward to show that if Γ1, Γ2 ∈ B0(**R**4), Γ1 ∩ Γ2 = ∅ then

$$\begin{aligned} \int \chi\_{\Gamma\_1 \cup \Gamma\_2} (p+q) \, \Omega\_{\mathfrak{m}} (dp) \, \Omega\_{\mathfrak{m}} (dq) &= \int \chi\_{\Gamma\_1} (p+q) \, \Omega\_{\mathfrak{m}} (dp) \, \Omega\_{\mathfrak{m}} (dq) + \\ &\int \chi\_{\Gamma\_2} (p+q) \, \Omega\_{\mathfrak{m}} (dp) \, \Omega\_{\mathfrak{m}} (dq) .\end{aligned}$$

Therefore

$$\begin{aligned} &\int \chi\_{\Gamma}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}}(dq) - \int \chi\_{\Gamma'}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}}(dq) \, | \, \le \\ &\int \chi\_{\Gamma'^{-}}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}}(dq) + \int \chi\_{\Gamma'^{+}}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}}(dq) .\end{aligned}$$

We will show that

$$\lim\_{\epsilon \to 0} (\epsilon^{-3} \int \chi\_{\Gamma'^{\pm}}(p+q) \, \Omega\_m(dp) \Omega\_m(dq)) = 0. \tag{62}$$

It suffices to consider the − case. We have

$$\begin{split} \int \chi\_{\Gamma^{-}}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \Omega\_{\mathfrak{m}}(dq) &\leq \int \chi\_{(a,a^{+}) \times B\_{\mathfrak{e}}(0)}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}}(dq) \\ &= \int \chi\_{(a,a^{+})}(\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{p}) + \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{q})) \chi\_{\begin{subarray}{c} R\_{\mathfrak{e}}(0) \stackrel{\rightarrow}{\to} \text{ $\bar{q}$ } \\ \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{q}) \end{subarray}}(\stackrel{\rightarrow}{p}) \, \frac{d\stackrel{\rightarrow}{p}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{p})} \\ \end{split} \tag{63}$$

We will come back to this equation later but will now return to the general argument Argument 1 and consider the second and final ≈. This ≈ arises because we are approximating - *p* by −- *q* since - *p* ranges over a ball of radius centred on −- *q* . ---

Suppose that - *p* and - *q* are such that *χ B*(0)− - *q* ( *p* ) = 1. Then | *p* + - *q* | < . Thus || *p* |−|- *q* || < .

Hence

$$\begin{array}{rcl} |\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{p}) - \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{q})| &=& |(\stackrel{\rightarrow}{p}^{2} + m^{2})^{\frac{1}{2}} - (\stackrel{\rightarrow}{q}^{2} + m^{2})^{\frac{1}{2}}| \\ &=& \left| \frac{\stackrel{\rightarrow}{p}^{2} - \stackrel{\rightarrow}{q}^{2}}{(\stackrel{\rightarrow}{p}^{2} + m^{2})^{\frac{1}{2}} + (\stackrel{\rightarrow}{q}^{2} + m^{2})^{\frac{1}{2}}} \right| \\ &\leq& \frac{\stackrel{\rightarrow}{p}^{2} - \stackrel{\rightarrow}{q}^{2}}{2m} \\ &=& \frac{|\stackrel{\rightarrow}{p}| - |\stackrel{\rightarrow}{q}| |(\stackrel{\rightarrow}{p}) + |\stackrel{\rightarrow}{q}|)}{2m} \\ &<& \frac{\epsilon}{2m} (|\stackrel{\rightarrow}{p}| + |\stackrel{\rightarrow}{q}|). \end{array}$$

We have | - *p* | ∈ (| - *q* | − , | - *q* | + ). Therefore | - *p* | + | - *q* | < 2| - *q* | + . Thus

$$|\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{p}) - \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{q})| < \frac{\epsilon}{2m}(2|\stackrel{\rightarrow}{q}| + \epsilon).$$

Therefore

$$\begin{split} \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}}) + \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}) &= \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}}) - \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}) + \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}) + \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}) \\ &\leq |\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}}) - \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})| + 2\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}) \\ &< 2\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}) + \frac{\mathfrak{c}}{2m}(2\|\stackrel{\rightarrow}{\mathcal{q}}\| + \mathfrak{c}). \end{split} \tag{64}$$

Now let

$$\begin{array}{rcl} I(\epsilon) & = & \int \chi\_{(a,b)}(\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}}) + \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})) \chi\_{\underset{B\_{\mathfrak{e}}(\stackrel{\rightarrow}{\mathcal{Q}}) - \stackrel{\rightarrow}{\mathcal{q}}}(\stackrel{\rightarrow}{\mathcal{P}})} \stackrel{\rightarrow}{\frac{d\stackrel{\rightarrow}{\mathcal{P}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}})}} \frac{d\stackrel{\rightarrow}{\mathcal{q}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}})} \\\ I(\epsilon) & = & \int \chi\_{(a,b)}(2\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})) \chi\_{\underset{B\_{\mathfrak{e}}(\stackrel{\rightarrow}{\mathcal{Q}}) - \stackrel{\rightarrow}{\mathcal{q}}}(\stackrel{\rightarrow}{\mathcal{p}})} \stackrel{\rightarrow}{\frac{d\stackrel{\rightarrow}{\mathcal{P}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{p}})}} \frac{d\stackrel{\rightarrow}{\mathcal{q}}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})} \\\ K(\epsilon) & = & \int \chi\_{(a,b)}(2\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})) \frac{\frac{4}{3}\pi\epsilon^{3}}{\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}})^{2}} d\stackrel{\rightarrow}{\mathcal{q}}. \end{array}$$

We will show that

$$\lim\_{\varepsilon \to 0} \varepsilon^{-\frac{3}{4}} (I(\varepsilon) - J(\varepsilon)) = 0, \text{ and } \lim\_{\varepsilon \to 0} \varepsilon^{-\frac{3}{4}} (f(\varepsilon) - K(\varepsilon)) = 0. \tag{65}$$

Concerning the first limit we note that *χ*(*a*,*b*)(*ωm*( - *p* ) + *ωm*( - *q* )) differs from *χ*(*a*,*b*)(2*ωm*( - *q* )) if and only if

Thus

$$|I(\mathfrak{e}) - J(\mathfrak{e})| = I\_1(\mathfrak{e}) + I\_2(\mathfrak{e}) + I\_3(\mathfrak{e}) + I\_4(\mathfrak{e}),\tag{66}$$

where

$$\begin{split} I\_{1}(\boldsymbol{\varepsilon}) &= \int \chi\_{\left(\boldsymbol{a},\boldsymbol{b}\right)} \left(\omega\_{m}(\stackrel{\dashrightarrow}{p}) + \omega\_{m}(\stackrel{\dashrightarrow}{q})\right) \chi\_{\left(-\infty,\boldsymbol{a}\right]} \left(2\omega\_{m}(\stackrel{\dashrightarrow}{q})\right) \chi\_{\underset{\stackrel{\dashrightarrow}{B\_{\epsilon}\left(\stackrel{\dashrightarrow}{\boldsymbol{0}}\right)}}{\operatorname{\small{\small}}}(\stackrel{\dashrightarrow}{p})} \\ & \xrightarrow{\stackrel{\dashrightarrow}{d}\stackrel{\dashrightarrow}{p}} \frac{\stackrel{\dashrightarrow}{q}}{\omega\_{m}(\stackrel{\dashrightarrow}{p})} ,\end{split} \tag{67}$$

and *I*2, *I*3, *I*4 are defined similarly. We will show that

$$\lim\_{\epsilon \to 0} \epsilon^{-\S} I\_1(\epsilon) = 0. \tag{68}$$

*I*2, *I*3 and *I*4 can be dealt with similarly.

Using Equation (64)

$$
\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{P}}) + \omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}) \in (a, b) \text{ and } 2\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}) \le a \Rightarrow a - \frac{\mathfrak{e}}{2m}(2|\stackrel{\rightarrow}{\mathcal{q}}| + \mathfrak{e}) < 2\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{\mathcal{q}}) \le a.
$$

Therefore

$$\begin{split} \mathcal{I}\_{1}(\boldsymbol{\epsilon}) &\leq \quad \int \chi\_{\left(\boldsymbol{a}-\left(\boldsymbol{2}\right)\overset{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{\Psi}}\right)\left(\boldsymbol{+}\boldsymbol{\epsilon}\right)\left(\boldsymbol{2}m\right),\boldsymbol{a}} \left(2\omega\_{\boldsymbol{m}}\left(\stackrel{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{\Psi}}\right)\right) \chi\_{\overset{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{\Psi}}\left(\stackrel{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{\Psi}}\right)\left(\stackrel{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{m}^{2}}\right)} \stackrel{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{\epsilon}} \stackrel{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{\Psi}} \, d\boldsymbol{\overline{\boldsymbol{p}}} \, d\boldsymbol{\overline{\boldsymbol{q}}} \\ &= \quad \, \, ^{4}\boldsymbol{\pi}\boldsymbol{\epsilon}^{3} \int \chi\_{\left(\boldsymbol{a}-\left(\boldsymbol{2}\right)\overset{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{\Psi}}\right)\left(\stackrel{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{\Psi}}\right)\left(\stackrel{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{m}^{2}}\right)\left(\stackrel{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{m}^{2}}\right) \stackrel{\textstyle \boldsymbol{\epsilon}}{\boldsymbol{m}^{2}} \, d\boldsymbol{\overline{\boldsymbol{q}}}.\end{split}$$

Hence

$$\left(\mathfrak{c}^{-3}I\_{1}(\mathfrak{c})\right)^{\*} \leq \frac{4}{3}\pi \int \chi\_{\begin{subarray}{c}(a-(2)\stackrel{\rightarrow}{q}|+c)\mathfrak{c}/(2m),a\end{subarray}}(2\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{q})) \frac{1}{m^{2}}d\,\stackrel{\rightarrow}{q}.\tag{69}$$

The integrand is integrable for all > 0, vanishes outside the compact set

$$\mathbb{C} = \{ \stackrel{\rightarrow}{q} \in \mathbb{R}^3 : 2\omega\_m(\stackrel{\rightarrow}{q}) \le a \},$$

is dominated by the integrable function

$$g(\stackrel{\rightharpoonup}{q}) = \frac{1}{m^2} \chi\_{[0,a]}(2\omega\_m(\stackrel{\rightharpoonup}{q}))\_{\checkmark}$$

and converges pointwise to 0 everywhere on **R**3 as → 0 except on the set *∂C* = { - *q* ∈ **R**3 : 2*ωm*( - *q* ) = *a*} which is a set of measure 0. Therefore by the dominated convergence theeorem

$$\lim\_{\epsilon \to 0} \epsilon^{-3} I\_1(\epsilon) = 0,\tag{70}$$

as required.

Now regarding the second limit in Equation (65) consider the function *f* : [0, ∞) → (0, *m*−1] defined by

$$f(p) = (m^2 + p^2)^{-\frac{1}{2}}.\tag{71}$$

*f* is analytic. Therefore by Taylor's theorem for all *q*, *p* ≥ 0

$$f(p) = f(q) + f'(q)(p - q) + \frac{1}{2}f''(\xi)(p - q)^2,\tag{72}$$

for some *ξ* between *q* and *p*. Now

$$\begin{aligned} f'(p) &= -p(m^2 + p^2)^{-\frac{3}{2}} \\ f''(p) &= (m^2 + p^2)^{-\frac{5}{2}}(2p^2 - m^2) \end{aligned}$$

Therefore

$$\begin{aligned} |f^{\prime\prime}(\xi)| &= (m^2 + \xi^2)^{-\frac{5}{2}} |2\xi^2 - m^2| \\ &\le m^{-5} (2(q + \epsilon)^2 + m^2) \end{aligned}$$

as long as |*p* − *q*| < . Thus

$$\begin{aligned} |f(p) - f(q)| &= |f'(q)(p - q) + \frac{1}{2}f''(\xi)(p - q)^2| \\ &< q(m^2 + q^2)^{-\frac{3}{2}}\epsilon + \frac{1}{2}m^{-5}(2(q + \epsilon)^2 + m^2)\epsilon^2 \\ &< m^{-1}\epsilon + \frac{1}{2}m^{-5}(2(q + \epsilon)^2 + m^2)\epsilon^2, \end{aligned}$$

as long as |*p* − *q*| < . Hence

$$\begin{split} |\left|f(\boldsymbol{\epsilon})-\boldsymbol{K}(\boldsymbol{\epsilon})| &= |\int \chi\_{(a,b)}(2\omega\_{m}(\stackrel{\scriptstyle\rightarrow}{q}))(\int \chi\_{\boldsymbol{B}\_{\boldsymbol{\epsilon}}(\stackrel{\scriptstyle\rightarrow}{q})-\stackrel{\scriptstyle\leftarrow}{q}}(\stackrel{\scriptstyle\rightarrow}{p})(\frac{1}{\omega\_{m}(\stackrel{\scriptstyle\rightarrow}{q})}-\frac{1}{\omega\_{m}(\stackrel{\scriptstyle\rightarrow}{q})}) \, d\stackrel{\scriptstyle\overline{\boldsymbol{p}}}{\underset{\scriptstyle\omega\_{m}(\stackrel{\scriptstyle\rightarrow}{q})}{\omega\_{m}(\stackrel{\scriptstyle\rightarrow}{q})}}| \\ &\leq \int \chi\_{(a,b)}(2\omega\_{m}(\stackrel{\scriptstyle\rightarrow}{q}))(\int \chi\_{\boldsymbol{B}\_{\boldsymbol{\epsilon}}(\stackrel{\scriptstyle\rightarrow}{0})-\stackrel{\scriptstyle\leftarrow}{q}}(\stackrel{\scriptstyle\overline{\boldsymbol{p}}})(|f(\stackrel{\scriptstyle\!\stackrel{\scriptstyle\!}{p})|-f(\stackrel{\scriptstyle\rightarrow}{q})|)}(\stackrel{\scriptstyle\overline{\boldsymbol{p}}}{\stackrel{\scriptstyle\rightarrow}{q}})) \\ &\leq \int \chi\_{(a,b)}(2\omega\_{m}(\stackrel{\scriptstyle\rightarrow}{q})) \int \chi\_{\boldsymbol{B}\_{\boldsymbol{\epsilon}}(\stackrel{\scriptstyle\rightarrow}{0})-\stackrel{\scriptstyle\leftarrow}{q}}(\stackrel{\scriptstyle\rightarrow}{p})(m^{-1}\boldsymbol{\epsilon}+\frac{1}{2}m^{-5}(2(|\stackrel{\scriptstyle\!\stackrel{\scriptstyle$$

Therefore

$$\begin{split} \lim\_{\epsilon \to 0} \epsilon^{-3} |I(\epsilon) - K(\epsilon)| &= \lim\_{\epsilon \to 0} \frac{4}{3} \pi \int \chi\_{\epsilon}(a, b) (2\omega\_{m}(\stackrel{\rightarrow}{q})) (m^{-1}\epsilon + \frac{1}{2}m^{-5}(2(|\stackrel{\rightarrow}{q}| + \epsilon)^{2} + m^{2})\epsilon^{2}) \\ \qquad \qquad \qquad \qquad \qquad \qquad \frac{\stackrel{\rightarrow}{d \, q}}{\omega\_{m}(\stackrel{\rightarrow}{q})} \\ &= 0, \end{split}$$

as required. We have therefore dealt with the second ≈ in Argument 1.

To finish dealing with the first ≈ suppose that 1 > 0 is given. Choose *c* ∈ (*a*, *b*) such that

$$\frac{16}{3}\pi^2(Z(\varepsilon) - Z(a)) < \frac{\epsilon\_1}{2}.\tag{73}$$

Now choose *δ*1 > 0 such that if 0 < < *δ*1 then

$$\begin{split} &|\boldsymbol{\varepsilon}^{-3} \int \chi\_{(\boldsymbol{a},\boldsymbol{c})}(\omega\_{\boldsymbol{m}}(\stackrel{\r\smile}{p}) + \omega\_{\boldsymbol{m}}(\stackrel{\r\smile}{q})) \chi\_{\underset{\boldsymbol{B}\_{\boldsymbol{\epsilon}}(\stackrel{\r\smile}{q})}{\operatorname{\mkern}}{\operatorname{\mkern}}{\operatorname{\mkern}}{\operatorname{\mkern}}\_{\operatorname{\mkern}}(\stackrel{\r\smile}{q})} (\stackrel{\r\smile}{\operatorname{\mkern}}{\operatorname{\mkern}}\_{\operatorname{\mkern}}(\stackrel{\r\smile}{q})) \chi\_{\underset{\boldsymbol{m}}(\stackrel{\r\smile}{q})}(\stackrel{\r\smile}{p}) \frac{d\stackrel{\r\smile}{p}}{\operatorname{\mkern}}\_{\operatorname{\mkern}}(\stackrel{\r\smile}{p}) \frac{d\stackrel{\r\smile}{q}}{\operatorname{\mkern}}\_{\operatorname{\mkern}}(\stackrel{\r\smile}{q}) \\ &\frac{4}{3}\pi \int \chi\_{(\boldsymbol{a},\boldsymbol{c})}(2\omega\_{\boldsymbol{m}}(\stackrel{\r\smile}{q})) \frac{d\stackrel{\r\smile}{q}}{\omega\_{\boldsymbol{m}}(\stackrel{\r\smile}{q})^{2}} | < \frac{\epsilon\_{1}}{2}. \end{split}$$

(We can do this because of the validity of the second ≈.) Choose *δ*2 > 0 such that if 0 < < *δ*2 then *a*+ = *a*+() < *c*. Let *δ* = min(*δ*1, *δ*2).

Then if < *δ* then

|−3 *χ*(*a*,*a*+)(*ωm*( - *p* ) + *ωm*( - *q* ))*χ B*( - 0 )− - *q* ( - *p* ) *d* - *p ωm*( - *p* ) *d* - *q ωm*( - *q* ) | ≤ |−3 *χ*(*a*,*c*)(*ωm*( - *p* ) + *ωm*( - *q* ))*χ B*( - 0 )− - *q* ( - *p* ) *d* - *p ωm*( - *p* ) *d* - *q ωm*( - *q* ) | ≤ |−3 *χ*(*a*,*c*)(*ωm*( - *p* ) + *ωm*( - *q* ))*χ B*( - 0 )− - *q* ( - *p* ) *d* - *p ωm*( - *p* ) *d* - *q ωm*( - *q* ) − 4 3 *π χ*(*a*,*c*)(2*ωm*( - *q* )) *d* - *q ωm*( - *q* )2 | + | 4 3 *π χ*(*a*,*c*)(2*ωm*( - *q* )) *d* - *q ωm*( - *q* )2 | < 1 2 + 4 3 *π χ*(*a*,*c*)(2*ωm*( - *q* )) *d* - *q ωm*( - *q* )2 = 1 2 + 4 3 *π Z*(*c*) *Z*(*a*) 4*πr*2 *m*2 + *r*2 *dr* ≤ 1 2 + 16 3 *π*2(*Z*(*c*) − *Z*(*a*)) < 1 2 + 1 2 = 1.

Thus

$$\lim\_{\varepsilon \to 0} \varepsilon^{-3} \int \chi\_{(a,\pi^{+})} (\omega\_{m}(\stackrel{\rightarrow}{p}) + \omega\_{m}(\stackrel{\rightarrow}{q})) \chi\_{\underset{B\_{\varepsilon}(\stackrel{\rightarrow}{0}) - \stackrel{\rightarrow}{q}}{\longrightarrow}}(\stackrel{\rightarrow}{p}) \frac{d\stackrel{\rightarrow}{p}}{\omega\_{m}(\stackrel{\rightarrow}{p})} \frac{d\stackrel{\rightarrow}{q}}{\omega\_{m}(\stackrel{\rightarrow}{q})} = 0,\tag{74}$$

thereby completing the proof of the validity of the first ≈ and therefore the validity of Argument 1.

#### **7. Investigation of the Measure Defined by the Convolution Ω***im ∗* **Ω***im*

The measure Ω+ *im* is defined by

$$\Omega\_{\rm im}^{+}(\Gamma) = \int\_{\pi(\Gamma \cap H\_{\rm im}^{+})} \frac{d\stackrel{\rightarrow}{\mathcal{P}}}{\omega\_{\rm im}(\stackrel{\rightarrow}{\mathcal{P}})} \text{ for } \Gamma \in \mathcal{B}\_{0}(\mathbf{R}^{4}), \tag{75}$$

where

$$H\_{\rm im}^{+} = \{ p \in \mathbb{R}^{4} : p^{2} = -m^{2}, p^{0} \ge 0 \}. \tag{76}$$

Ω+ *im* is a measure concentrated on the positive time imaginary mass hyperboloid *H*+ *im* corresponding to mass *im*. There is also a measure Ω− *im* on *H*− *im* and we may define Ω*im* = Ω+ *im* + Ω− *im*, for *m* > 0. Ω*im* is a Lorentz invariant measure on *Him* = {*p* ∈ **R**4 : *p*2 = −*m*2}.

Define, for *m* ∈ **C**

$$J\_m^+ = \{ p \in \mathbb{C}^4 : p^2 = m^2, \text{Re}(p^0) \ge 0, \text{Im}(p^0) \ge 0 \},\tag{77}$$

where *p*2 = *ημν pμ pν* (in which *ημν* is the Minkowski space metric tensor). Then, for *m* > 0,

$$f\_m^+ \cap \mathbb{R}^4 = \{ p \in \mathbb{R}^4 : p^2 = m^2, p^0 \ge 0 \} = H\_{m'}^+ \tag{78}$$

$$\begin{aligned} \left| f\_{\text{in}}^{+} \cap (i\mathbb{R}^{4}) \right| &= \left\{ p \in i\mathbb{R}^{4} : p^{2} = m^{2}, \text{Re}(p^{0}) \ge 0, \text{Im}(p^{0}) \ge 0 \right\} \\ &= \left\{ iq : q \in \mathbb{R}^{4}, q^{2} = -m^{2}, q^{0} \ge 0 \right\} \\ &= \left\{ iH\_{im}^{+} \right. \end{aligned} \tag{79}$$

Now if - *p* ∈ **R**3, *m* > 0, we may write

$$\omega\_{\scriptscriptstyle \rm im}(\stackrel{\rightharpoonup}{p}) = ((\textit{im})^2 + \stackrel{\rightharpoonup}{p}^2)^{\frac{1}{2}} = (-\textit{m}^2 + \stackrel{\rightharpoonup}{p}^2)^{\frac{1}{2}} = (-(\textit{m}^2 + (\stackrel{\rightharpoonup}{i}\stackrel{\rightharpoonup}{p})^2))^{\frac{1}{2}} = i(\textit{m}^2 + (\stackrel{\rightharpoonup}{i}\stackrel{\rightharpoonup}{p})^2)^{\frac{1}{2}} = i\omega\_{\textit{m}}(\stackrel{\rightharpoonup}{i}\stackrel{\rightharpoonup}{p}).$$

One may consider the measure Ω+ *m* to be defined on *i***R**4 as well as **R**4 and for all *m* ∈ **R** or *m* ∈ *i***R** according to

$$
\Omega\_m^+ (\Gamma) = \int\_{\pi(\Gamma \cap J\_m^+)} \frac{d\stackrel{\rightarrow}{p}}{\omega\_m(\stackrel{\rightarrow}{p})}.\tag{80}
$$

Then from Equation (79)

$$
\Omega\_m^+(i\Gamma) = \int\_{i\pi(\Gamma \cap H\_{im}^+)} \frac{d\stackrel{\rightarrow}{p}}{\omega\_m(\stackrel{\rightarrow}{p})}.\tag{81}
$$

Now make the substitution - *p* = *i* - *q* . Then *d* - *p* = −*id*- *q* . Thus

$$
\Omega\_m^+(i\Gamma) = \int\_{\pi(\Gamma \cap H\_{im}^+)} \frac{-i d\stackrel{\rightarrow}{\vec{q}}}{-i\omega\_{im}(\stackrel{\rightarrow}{\vec{q}})} = \Omega\_{im}^+(\Gamma). \tag{82}
$$

Now suppose that

$$
\psi = \sum\_{k} c\_{k} \chi\_{E\_{k'}} \tag{83}
$$

where *ci* ∈ **C** and *Ek* ∈ B0(**R**4), is a simple function. Then

$$\begin{split} \int\_{\mathbf{R}^4} \psi(p) \, \Omega\_{im}^+(dp) &= \sum\_k c\_k \Omega\_{im}^+(E\_k) \\ &= \sum\_k c\_k \Omega\_{m}^+(iE\_k) \\ &= \sum\_k c\_k \int\_{i\mathbf{R}^4} \chi\_{iE\_k}(p) \, \Omega\_{m}^+(dp) \\ &= \sum\_k c\_k \int\_{i\mathbf{R}^4} \chi\_{E\_k}\left(\frac{p}{i}\right) \Omega\_{m}^+(dp) \\ &= \int\_{i\mathbf{R}^4} \psi\left(\frac{p}{i}\right) \Omega\_{m}^+(dp) .\end{split} \tag{84}$$

As this is true for every such simple function *ψ* it follows that

$$
\int\_{\mathbf{R}^4} \psi(p) \, \Omega\_{im}^+(dp) = \int\_{i\mathbf{R}^4} \psi(\frac{p}{i}) \, \Omega\_m^+(dp), \tag{85}
$$

for every function *ψ* which is integrable with respect to Ω+ *im*. Therefore

$$\begin{split} (\Omega\_{im}^{+} \* \Omega\_{im}^{+})(\Gamma) &= \int\_{(\mathbb{R}^{4})^{2}} \chi \boldsymbol{r} (\boldsymbol{p} + \boldsymbol{q}) \, \Omega\_{im}^{+} (d\boldsymbol{p}) \, \Omega\_{im}^{+} (d\boldsymbol{q}) \\ &= \int\_{(i\mathbb{R}^{4})^{2}} \chi \boldsymbol{r} \left(\frac{\boldsymbol{p} + \boldsymbol{q}}{i}\right) \, \Omega\_{m}^{+} (d\boldsymbol{p}) \, \Omega\_{m}^{+} (d\boldsymbol{q}) \\ &= \int\_{(i\mathbb{R}^{4})^{2}} \chi \boldsymbol{r} (\boldsymbol{p} + \boldsymbol{q}) \Omega\_{m}^{+} (d\boldsymbol{p}) \Omega\_{m}^{+} (d\boldsymbol{q}) \\ &= (\Omega\_{m}^{+} \* \Omega\_{m}^{+}) (i\Gamma) .\end{split} \tag{86}$$

for all Γ ∈ B0(**R**4).

Now in general, suppose that a measure *μ* has a causal spectral representation of the form

$$\mu(\Gamma) = \int\_{m'=0}^{\infty} \Omega\_{m'}^+(\Gamma) \,\sigma(m'),\tag{87}$$

for some Borel spectral measure *σ* : B0([0, ∞)) → **C**. Then *μ* extends to a measure defined on *i***R**4 by

$$
\mu(i\Gamma) = \int\_{m'=0}^{\infty} \Omega\_{m'}^+(i\Gamma) \,\sigma(dm') = \int\_{m'=0}^{\infty} \Omega\_{im'}^+(\Gamma) \,\sigma(dm'),\tag{88}
$$

for Γ ∈ B0(**R**4). Therefore since, as we have determined above, Ω+ *m* ∗ Ω+ *m* is a causal spectral measure with spectrum

$$\sigma(m') = \begin{cases} 4\pi m Z(m') \text{ for } m' \ge 2m \\ 0 \text{ otherwise,} \end{cases} \tag{89}$$

it follows that

$$(\Omega\_m^+ \ast \Omega\_m^+)(i\Gamma) = \int\_{m'=0}^\infty \Omega\_{im'}^+(\Gamma) \,\sigma(dm'). \tag{90}$$

Therefore using Equation (86) Ω+ *im* ∗ Ω+ *im* is a measure with spectral representation

$$(\Omega\_{im}^+ \ast \Omega\_{im}^+) (\Gamma) = \int\_{m'=0}^\infty \Omega\_{im'}^+ (\Gamma) \,\sigma(m') \,dm',\tag{91}$$

where *σ* is the spectral function given by Equation (89). Note that Ω+ *im* ∗ Ω+ *im* is not causal, it is a type III measure, and

$$\text{supp}(\Omega\_{\text{im}}^+ \ast \Omega\_{\text{im}}^+) = \{ p \in \mathbb{R}^4 : p^2 \le -4m^2, p^0 \ge 0 \}. \tag{92}$$

#### **8. Determination of the Density Defining a Causal Lorentz Invariant Borel Measure from Its Spectrum**

Suppose that *μ* is of the form of Equation (30) where *σ* is a well behaved (e.g., locally integrable) function. We would like to see if *μ* can be defined by a density with respect to the Lebesgue measure, i.e., if there exists a function *g* : **R**4 → **C** such that

$$
\mu(\Gamma) = \int\_{\Gamma} \lg(p) \, dp. \tag{93}
$$

Well we have that

$$\mu(\Gamma) = \int\_{m=0}^{\infty} \sigma(m) \Omega\_m^+(\Gamma) \, dm = \int\_{m=0}^{\infty} \sigma(m) \int\_{\pi(\Gamma \cap H\_m^+)} \frac{d\overline{p}}{\omega\_m(\overline{\mathcal{P}})} \, dm. \tag{94}$$

Now

$$\begin{array}{rcl} \stackrel{\rightharpoonup}{p} \in \pi(\Gamma \cap H\_{\text{m}}^{+}) & \Leftrightarrow & (\exists p \in \mathbb{R}^{4}) \stackrel{\rightharpoonup}{p} = \pi(p), p \in H\_{\text{m}}^{+}, p \in \Gamma\\ & & \Leftrightarrow & (\omega\_{\text{m}}(\stackrel{\rightharpoonup}{p}), \stackrel{\rightharpoonup}{p}) \in \Gamma\\ & & \Leftrightarrow & \chi\_{\Gamma}(\stackrel{\rightharpoonup}{\omega\_{\text{m}}(\stackrel{\rightharpoonup}{p})}, \stackrel{\rightharpoonup}{p}) = 1. \end{array}$$

Therefore

$$
\mu(\Gamma) = \int\_{m=0}^{\infty} \sigma(m) \int\_{\mathbb{R}^3} \chi\_{\Gamma}(\omega\_m(\stackrel{\rightarrow}{\mathcal{P}}), \stackrel{\rightarrow}{\mathcal{P}}) \frac{1}{\omega\_m(\stackrel{\rightarrow}{\mathcal{P}})} \, d\stackrel{\rightarrow}{\mathcal{P}} \, dm. \tag{95}
$$

Now consider the transformation defined by the function *h* : (0, ∞) × **R**3 → **R**4 given by

$$h(m, \stackrel{\rightarrow}{p}) = (\omega\_m(\stackrel{\rightarrow}{p}), \stackrel{\rightarrow}{p}).\tag{96}$$

Let

$$q = h(m, \stackrel{\rightharpoonup}{p}) = (\omega\_{\mathfrak{m}}(\stackrel{\rightharpoonup}{p}), \stackrel{\rightharpoonup}{p}) = ((m^2 + \stackrel{\rightharpoonup}{p}^2)^{\frac{1}{2}}, \stackrel{\rightharpoonup}{p}).\tag{97}$$

Then

$$\frac{\partial q^0}{\partial m} = m\omega\_m(\stackrel{\rightarrow}{p})^{-1}, \frac{\partial q^0}{\partial p^j} = p^j \omega\_m(\stackrel{\rightarrow}{p})^{-1}, \frac{\partial q^i}{\partial m} = 0, \frac{\partial q^i}{\partial p^j} = \delta\_{ij}.\tag{98}$$

for *i*, *j* = 1, 2, 3. Thus the Jacobian of the transformation is

$$J(m, \stackrel{\rightarrow}{p}) = m\omega\_{\mathfrak{m}}(\stackrel{\rightarrow}{p})^{-1}.\tag{99}$$

Now *q* = (*ωm*( - *p* ), - *p* ). Therefore *q*2 = *ωm*( - *p* )2 − - *p* 2 = *m*2. So *m* = (*q*2) 1 2 , *q*2 > 0. Thus

$$\begin{split} \mu(\Gamma) &= \int\_{q \in \mathbb{R}^4, q^2 > 0, q^0 > 0} \chi\_\Gamma(q) \frac{\sigma(m)}{\omega\_\mathfrak{m}(\stackrel{\rightarrow}{\mathcal{P}})} \frac{dq}{f(m, \stackrel{\rightarrow}{\mathcal{P}})} \\ &= \int\_{q^2 > 0, q^0 > 0} \chi\_\Gamma(q) \frac{\sigma(m)}{m} \, dq. \end{split} \tag{100}$$

Hence

$$\begin{aligned} \mu(\Gamma) &= \int\_{q^2 > 0, q^0 > 0} \chi \mathbf{r} \,(q) \frac{\sigma((q^2)^{\frac{1}{2}})}{(q^2)^{\frac{1}{2}}} \, dq \\ &= \quad \int\_{\Gamma} \mathbf{g}(q) \, dq\_{\prime} \end{aligned}$$

where *g* : **R**4 → **C** is defined by

$$\mathcal{g}(q) = \begin{cases} \ (q^2)^{-\frac{1}{2}} \sigma((q^2)^{\frac{1}{2}}) \text{ if } q^2 > 0, q^0 > 0\\ 0 \text{ otherwise.} \end{cases} \tag{101}$$

We have therefore shown how, given a spectral representation of a causal Lorentz invariant Borel complex measure in which the spectrum is a complex function, one can obtain and equivalent representation of the measure in terms of a density with respect to Lebesgue measure.

#### **9. Convolutions and Products of Causal Lorentz Invariant Borel Measures**

#### *9.1. Convolution of Measures*

Let *μ* and *ν* be causal Lorentz invariant Borel complex measures. Then (up to possible atoms at the origin which can be dealt with in a straightforward way) there exist Borel spectral measures *σ*, *ρ* : B0([0, ∞)) → **C** such that

$$\begin{aligned} \mu &= \int\_{m=0}^{\infty} \Omega\_m \, \sigma(dm), \\ \nu &= \int\_{m=0}^{\infty} \Omega\_m \, \rho(dm). \end{aligned} \tag{102}$$

We will assume, without loss of generality, that *σ* and *ρ* are complex measures, i.e. *σ*, *ρ* : B([0, ∞)) → **C** and are countably additive. The convolution of *μ* and *ν*, if it exists, is given by

$$(\mu \* \nu)(\Gamma) = \int \chi\_{\Gamma}(p+q)\,\mu(dp)\,\nu(dq). \tag{103}$$

Now let *ψ* = ∑*i ciχEi* with *ci* ∈ **C**, *Ei* ∈ B0(**R**4) be a simple function. Then

$$\begin{split} \int \Psi(p)\,\mu(dp) &= \quad \int \sum\_{i} c\_{i} \chi\_{E\_{i}} \,\mu(dp) \\ &= \quad \sum\_{i} c\_{i} \mu(E\_{i}) \\ &= \quad \sum\_{i} c\_{i} \int\_{m=0}^{\infty} \Omega\_{\mathfrak{m}}(E\_{i}) \,\sigma(dm) \\ &= \quad \sum\_{i} c\_{i} \int\_{m=0}^{\infty} \int\_{\mathbf{R}^{4}} \chi\_{E\_{i}}(p) \,\Omega\_{\mathfrak{m}}(dp) \,\sigma(dm) \\ &= \quad \int\_{\mathfrak{m}=0}^{\infty} \int\_{\mathbf{R}^{4}} \psi(p) \,\Omega\_{\mathfrak{m}}(dp) \,\sigma(dm) .\end{split}$$

Therefore for any sufficiently well behaved measurable function *ψ* : **R**4 → **C** (e.g. bounded measurable functions of compact support)

$$
\int \psi(p)\mu(dp) = \int \psi(p)\,\Omega\_{\mathfrak{M}}(dp)\,\sigma(dm).\tag{104}
$$

(Note that the integral exists because *σ* is a Borel measure.) Hence for all Γ ∈ B0(**R**4)

$$\begin{split} \chi(\mu \ast \nu)(\Gamma) &= \int \chi\_{\Gamma}(p+q) \, \mu(dp) \, \nu(dq) \\ &= \int \chi\_{\Gamma}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \sigma(dm) \, \Omega\_{\mathfrak{m}'}(dq) \, \rho(dm') \\ &= \int \chi\_{\Gamma}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}'}(dq) \, \sigma(dm) \, \rho(dm'), \end{split} \tag{105}$$

by Fubini's theorem, as long as

$$\int \chi\_{\Gamma}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}'}(dq) |\sigma|(dm) < \infty, \forall m' \in [0, \infty), \tag{106}$$

where |*σ*| is the total variations of the measure *σ*.

Suppose that Γ ∈ B0(**R**4). Then there exists *a*, *R* ∈ (0, ∞) such that Γ ⊂ (−*a*, *a*) × *BR*( - 0 ), where *BR*( - 0 ) = { - *p* ∈ **R**3 : | - *p* | < *R*}. Now

$$\int \chi\_{\Gamma}(p+q)\,\Omega\_{\mathfrak{m}}(dp) = \int\_{\Gamma-q} \Omega\_{\mathfrak{m}}(dp) = \Omega\_{\mathfrak{m}}(\Gamma-q) < \infty,\tag{107}$$

for all *q* ∈ **R**4 because Ω*m* is Borel and Γ is compact.

Now suppose that *m*, *m* > *a*. Then

$$p \in H\_m^+, q \in H\_{m'}^+ \Rightarrow (p+q)^0 = p^0 + q^0 \ge m + m' > 2a \Rightarrow (p+q) \notin \Gamma. \tag{108}$$

Thus

$$
\int \chi\_{\Gamma}(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}'}(dq) = 0. \tag{109}
$$

Therefore since *σ* and *ρ* are Borel, (*μ* ∗ *ν*)(Γ) exists, is finite and is given by Equation (105). Now let Λ ∈ *O*(1, 3)+↑, *ψ* : **R**4 → **C** be a measurable function of compact support. Then

$$\begin{aligned} \left< \mu \ast \nu, \Lambda \psi > & \quad = \int \psi(\Lambda^{-1}(p+q)) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}'}(dq) \, \sigma(dm) \, \rho(dm') \right> \\ &= \int \psi(p+q) \, \Omega\_{\mathfrak{m}}(dp) \, \Omega\_{\mathfrak{m}'}(dq) \, \sigma(dm) \, \rho(dm'). \\ &= \quad <\mu \ast \nu, \psi > \end{aligned}$$

Therefore *μ* ∗ *ν* is Lorentz invariant. It can be shown, by an argument similar to that used for the case Ω*m* ∗ Ω*m* that *μ* ∗ *ν* is causal.

We have therefore shown that the convolution of two causal Lorentz invariant Borel complex measures exists and is a causal Lorentz invariant Borel complex measure.

#### *9.2. Product of measures*

We now turn to the problem of computing the product of two causal Lorentz invariant Borel complex measures. The problem of computing the product of measures or distributions is difficult in general and has attracted a large amount of research [10,19,20]. In such work one generally seeks a definition of the product of measures or distributions which agrees with the ordinary product when the measures or distributions are functions (i.e., densities with respect to Lebesgue measure). The most common approach is to use the fact that, for Schwartz functions *f* , *g* ∈ S(**R**4) multiplication in the spatial domain corresponds to convolution in the frequency domain, i.e., (*f g*)∧ = *f* ∧ ∗ *g*∧ (where ∧ denotes the Fourier transform operator). Thus one defines the product of measures or distributions *μ*, *ν* as

$$
\mu\nu = (\mu^\wedge \ast \nu^\wedge)^\vee. \tag{110}
$$

However, this definition is only successful when the convolution that it involves exists which may not be the case in general. If *μ*, *ν* are tempered measures then *μ*∧ and *ν*∧ exist as tempered distributions, however, they are generally not causal, even if *μ*, *ν* are causal.

We will therefore not use the "frequency space" approach to define the product of measures but will use a different approach. Our approach is just as valid as the frequency space approach because our product will coincide with the usual function product when the measures are defined by densities. Furthermore, our approach is useful for the requirements of QFT because measures and distributions in QFT are frequently Lorentz invariant and causal.

Let int(*C*) = {*p* ∈ **R**4 : *p*2 > 0, *p*0 > 0}. Suppose that *f* : int(*C*) → **C** is a Lorentz invariant locally integrable function. Then it defines a causal Lorentz invariant Borel measure *μf* which, by the spectral theorem, must have a representation of the form

$$\mu\_f(\Gamma) = \int\_{\Gamma} f(p) \, dp = \int\_{m=0}^{\infty} \Omega\_m(\Gamma) \, \sigma(dm),\tag{111}$$

for some spectral measure *σ* : B0([0, ∞)) → **C**. As *μf* is absolutely continuous with respect to Lebesgue measure it follows that *σ* must be non-singular, i.e., a function. By the result of the previous section a density defining *μf* is ∼ *f* : int(*C*) → **C** defined by

$$\widetilde{f}(p) = (p^2)^{-\frac{1}{2}} \sigma((p^2)^{\frac{1}{2}}), p \in \text{int}(\mathbb{C}).\tag{112}$$

We must have that ∼ *f* = *f* (almost everywhere). Therefore (almost everywhere on int(*C*))

$$f(p) = (p^2)^{-\frac{1}{2}} \sigma((p^2)^{\frac{1}{2}}).\tag{113}$$

Without loss of generality, it can be assumed that equality holds everywhere in Equation (113). *f*(*p*) depends only on *p*2. Therefore for all *m* > 0, *σ*(*m*) = *m f*(*p*) for all *p* ∈ int(*C*) such that *p*2 = *m*2. In particular

$$
\sigma(m) = m f((m, \stackrel{\rightarrow}{0})^T), \forall m > 0. \tag{114}
$$

Now we are seeking a definition of product which has useful properties. Two such properties would be that it is distributive with respect to generalized sums such as integrals and also that it agrees with the ordinary product when the measures are defined by functions. Suppose that we had such a product. Let *f* , *g* : int(*C*) → **C** be Lorentz invariant locallly integrable functions. Let *μ*, *ν* : B0(int(*C*)) → **C** be the associated measures with spectra *σ*, *ρ*. Then

$$\begin{array}{rcl} \mu\nu &=& \int\_{m=0}^{\infty} \Omega\_m \sigma(dm) \int\_{m'=0}^{\infty} \Omega\_{m'} \rho(dm')\\ &=& \int\_{m=0}^{\infty} \Omega\_m \, mf((m, \stackrel{\r}{0})^T) \, dm \int\_{m'=0}^{\infty} \Omega\_{m'} \, m' \, g((m', \stackrel{\r}{0})^T) \, dm'\\ &=& \int\_{m=0}^{\infty} \int\_{m'=0}^{\infty} \Omega\_m \Omega\_{m'} \, mf((m, \stackrel{\r}{0})^T) m' \, g((m', \stackrel{\r}{0})^T) \, dm \, dm'. \end{array}$$

Now we want this to be equal to

$$\int\_{m=0}^{\infty} \Omega\_m m(f\emptyset) ((m, \stackrel{\rightarrow}{0})^T) \, dm \tag{115}$$

This will be the case (formally) if we have

$$
\Omega\_{\rm m} \Omega\_{\rm m'} = \frac{1}{m} \delta(m - m') \Omega\_{\rm m}, \forall m, m' > 0. \tag{116}
$$

Physicists will be familiar with such a formula (e.g., the equal time commutation relations). Rather than attempting to define its meaning in a rigorous way, we will simply carry out the following formal computation for general Lorentz invariant Borel measures *μ*, *ν* with spectra *σ*, *ρ*

$$\begin{split} \mu\nu &= \quad \int\_{m=0}^{\infty} \Omega\_{\mathfrak{m}} \, \sigma(dm) \int\_{m'=0}^{\infty} \Omega\_{\mathfrak{m'}} \, \rho(dm') \\ &= \quad \int\_{m=0}^{\infty} \int\_{m'=0}^{\infty} \Omega\_{\mathfrak{m}} \Omega\_{\mathfrak{m'}} \sigma(m) \rho(m') \, dm \, dm' \\ &= \quad \int\_{m=0}^{\infty} \int\_{m'=0}^{\infty} \frac{1}{m} \Omega\_{\mathfrak{m}} \delta(m-m') \sigma(m) \rho(m') \, dm' \, dm \\ &= \quad \int\_{m=0}^{\infty} \frac{1}{m} \Omega\_{\mathfrak{m}} \sigma(m) \rho(m) \, dm. \end{split}$$

Therefore we can simply define the product *μν* in general by

$$
\mu\nu = \int\_{m=0}^{\infty} \frac{1}{m} \Omega\_m(\sigma\rho)(dm),
\tag{117}
$$

i.e.,

$$(\mu\nu)(\Gamma) = \int\_{m=0}^{\infty} \frac{1}{m!} \Omega\_{\mathfrak{m}}(\Gamma) \, (\sigma\rho)(dm),\tag{118}$$

for Γ ∈ B0(**R**4).

We have therefore reduced the problem of computing the product of measures on int(*C*) to the problem of computing the product of their 1D spectral measures. The problem of multiplying 1D measures is somewhat less problematic than the problem of multiplying 4D measures. A large class of 1D measures is made up of measures which are of the form of a function plus a finite number of "atoms" (singularities of the form *cδa* where *c* ∈ **C**\{0}, *a* ∈ [0, ∞), where *δa* is the Dirac delta function (measure) concentrated at *a*). There are other pathological types of the 1D measure but these may not be of interest for physical applications.

In the general non-pathological case, if *μ*, *ν* are causal Lorentz invariant Borel measures with spectra *σ*(*m*) = *ξ*(*m*) + ∑*k i*=1 *ciδ*(*m* − *ai*), *ρ*(*m*) = *ζ*(*m*) + ∑*l j*=1 *djδ*(*m* − *bj*) where *ξ*, *ζ* : [0, ∞) → **C** are locally integrable functions, *ci*, *dj* ∈ **C**\{0}, *k*, *l* ≥ 0, *ai*, *bj* ∈ [0, ∞) are such that *ai* = *bj*, ∀*i*, *j* then we may define the product of *μ* and *ν* to be the causal Lorentz invariant measure *μν* given by

$$
\mu\nu = \int\_{m=0}^{\infty} \Omega\_m \tau(dm),
\tag{119}
$$

where

$$\begin{aligned} \tau(m) &= \frac{1}{m} (\prescript{\tau}{}{\zeta}(m)\prescript{\tau}{}{\zeta}(m) + \prescript{\tau}{}{\zeta}(m) \sum\_{i=1}^{k} c\_i \delta(m - a\_i) + \prescript{\tau}{}{\zeta}(m) \sum\_{j=1}^{l} d\_j \delta(m - b\_j)), \\ &= \frac{1}{m} (\prescript{\tau}{}{\zeta}(m)\prescript{\tau}{}{\zeta}(m) + \sum\_{i=1}^{k} \prescript{\tau}{}{\zeta}(a\_j) c\_i \delta(m - a\_i) + \sum\_{j=1}^{l} \prescript{\tau}{}{\zeta}(b\_j) d\_j \delta(m - b\_j)), \end{aligned}$$

for *m* > 0.

#### **10. Conclusions**

We have defined a spectral calculus that enables one to compute the spectrum of any causal Lorentz invariant Borel complex measure on Minkowski space whose spectrum is a continuous function. This calculus can be used in many applications in QFT and leads to a method called spectral regularization [21].

We have computed the spectra associated with certain elementary convolutions involving Feynman propagators of mass *m* scalar particles. It has been shown how one can compute the density associated with a causal Lorentz invariant Borel complex measure from its spectrum.

We have shown that the convolution of arbitrary measures of the prescribed type exists and how their product exists in a wide class of cases of physical interest. Methods for the computation of these objects from the spectra of their components have been presented.

The spectral calculus can be used to compute the spectrum, and hence density, associated with the contraction of the vacuum polarization tensor [21]. A generalization of the spectral calculus to Lorentz invariant tensor valued measures on Minkowski space can be used to compute the form of the vacuum polarization tensor and therefore to compute the vacuum polarization function. This function is shown to have a close agreement, up to finite renormalization, with the vacuum polarization function obtained using dimensional regularization /renormalization. This can be used to compute the Uehling potential function without using renormalization from which the Uehling contribution to the Lamb shift for the H atom can be computed exactly.

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

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**

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## *Article* **An Upper Bound of the Third Hankel Determinant for a Subclass of** *q***-Starlike Functions Associated with** *k***-Fibonacci Numbers**

#### **Muhammad Shafiq 1, Hari M. Srivastava 2,3,4, Nazar Khan 1,\*, Qazi Zahoor Ahmad 5, Maslina Darus 6 and Samiha Kiran 1**

Received: 15 May 2020; Accepted: 16 June 2020; Published: 22 June 2020

**Abstract:** In this paper, we use *q*-derivative operator to define a new class of *q*-starlike functions associated with *k*-Fibonacci numbers. This newly defined class is a subclass of class A of normalized analytic functions, where class A is invariant (or symmetric) under rotations. For this function class we obtain an upper bound of the third Hankel determinant.

**Keywords:** starlike functions; subordination; *q*-Differential operator; *k*-Fibonacci numbers

**MSC:** Primary 05A30, 30C45; Secondary 11B65, 47B38

#### **1. Introduction and Definitions**

The calculus without the notion of limits is called quantum calculus; it is usually called *q*-calculus or *q*-analysis. By applying *q*-calculus, univalent functions theory can be extended. Moreover, the *q*-derivative, such as the *q*-calculus operators (or the *q*-difference) operator, are used to developed a number of subclasses of analytic functions (see, for details, the survey-cum-expository review article by Srivastava [1]; see also a recent article [2] which appeared in this journal, *Symmetry*).

Ismail et al. [3] instigated the generalization of starlike functions by defining the class of *q*-starilke functions. A firm footing of the usage of the *q*-calculus in the context of Geometric Functions Theory was actually provided and the basic (or *q*-) hypergeometric functions were first used in Geometric Function Theory by Srivastava (see, for details [4]). Raghavendar and Swaminathan [5] studied certain basic concepts of close-to-convex functions. Janteng et al. [6] published a paper in which the (*q*) generalization of some subclasses of analytic functions have studied. Further, *q*-hypergeometric functions, the *q*-operators were studied in many recent works (see, for example, [7–9]). The *q*-calculus applications in operator theory could be found in [4,10]. The coefficient inequality for *q*-starlike and *q*-close-to-convex functions with respect to Janowski functions were studied by Srivastava et al. [8,11] recently, (see also [12]). Further development on this subject could be seen in [7,9,13,14]. For a

comprehensive review of the theory and applications of the *q*-derivative (or the *q*-difference) operator and related literature, we refer the reader to the above-mentioned work [1].

We denote by A the class of functions which are analytic and having the form:

$$f\left(z\right) = z + \sum\_{n=2}^{\infty} a\_n z^n \tag{1}$$

in the open unit disk U given by

$$\mathcal{U} = \{ z : z \in \mathbb{C} \quad \text{and} \quad |z| < 1 \}.$$

and normalized by the following conditions:

$$f
\left(0\right) = 0 = f'\left(0\right) - 1.$$

The subordinate between two functions *f* and *g* in U, given by:

$$f \preccurlyeq g \quad \text{or} \quad f\left(z\right) \preccurlyeq g\left(z\right) \cdot \newline$$

if an analytic Schwarz function *w* exists in such way that

$$w\left(0\right) = 0 \qquad \text{and} \qquad \left|w\left(z\right)\right| < 1\_{\prime}$$

so that

$$f\left(z\right) = \operatorname{g}\left(w\left(z\right)\right).$$

In particular, the following equivalence also holds for the univalent function *g*

$$f(z) \preccurlyeq g(z) \quad (z \in \mathbb{U}) \implies f(0) = \emptyset(0) \quad \text{and} \quad f(\mathbb{U}) \subset \operatorname{g}(\mathbb{U}).$$

Next by the P class of analytic functions, *p*(*z*) in U is denoted, in which normalization conditions are given as follow:

$$p\left(z\right) = 1 + \sum\_{n=1}^{\infty} c\_n z^n \tag{2}$$

such that

$$\Re\left(p\left(z\right)\right) > 0 \qquad \qquad \left(\forall \, z \in \mathbb{U}\right) \dots$$

Let *k* be any positive real number, then we define the *k*-Fibonacci number sequence {*Fk*,*n*}∞ *n*=0 recursively by

$$F\_{k,0} = 0, \; F\_{k,1} = 1 \quad \text{and} \; F\_{k,n+1} = kF\_{k,n} + F\_{k,n-1} \quad \text{for} \; n \ge 1. \tag{3}$$

The *nth k*-Fibonacci number is given by

$$\begin{aligned} F\_{k,n} &= \frac{(k - \overline{T\_k})^n - \overline{T\_k}^n}{\sqrt{k^2 + 4}}, \\\\ \overline{T\_k} &= \frac{k - \sqrt{k^2 + 4}}{2}. \end{aligned} \tag{4}$$

where

If

$$\bar{p}\_k(z) = 1 + \sum\_{n=1} \bar{p}\_{k,n} z^n \rho$$

then we have (see also [15])

$$
\mathfrak{p}\_{k,n} = (F\_{k,n-1} + F\_{k,n+1}) \mathcal{T}\_k^n \qquad (n \in \mathbb{N}; \ N := \{1, 2, 3, \dots\}).\tag{5}
$$

**Definition 1.** *Let q* ∈ (0, 1) *then the q-number* [*λ*]*q is given by*

$$[\lambda]\_q = \begin{cases} \frac{1 - q^{\lambda}}{1 - q} & (\lambda \in \mathbb{C}) \\\sum\_{k=0}^{n-1} q^k = 1 + q + q^2 + \dots + q^{n-1} & (\lambda = n \in \mathbb{N}) \end{cases}$$

**Definition 2.** *The q-difference (or the q-derivative)* D*q operator of any given function f is defined, in a given subset of* C, *of complex numbers by*

$$\left(\mathcal{D}\_{\emptyset}f\right)(z) = \begin{cases} \frac{f\left(z\right) - f\left(qz\right)}{\left(1 - q\right)z} & \quad (z \neq 0) \\\ f'\left(0\right) & \quad (z = 0) \end{cases}$$

*led to the existence of the derivative f* (0)*.*

From Definitions 1 and 2, we have

$$\lim\_{q \to 1-} \left(\mathcal{D}\_q f\right)(z) = \lim\_{q \to 1-} \frac{f\left(z\right) - f\left(qz\right)}{\left(1 - q\right)z} = f'\left(z\right).$$

for a differentiable function *f* . In addition, from (1) and (2), we observe that

$$\left(\mathcal{D}\_q f\right)(z) = 1 + \sum\_{n=2}^{\infty} \left[n\right]\_q a\_n z^{n-1}.\tag{6}$$

In the year 1976, it was Noonan and Thomas [16] who concentrated on the function *f* given in (1) and gave the qth Hankel determinant as follows.

Let *n* 0 and q ∈ N. Than the qth Hankel determinant is defined by

Several authors studied the determinant *H*q (*n*). In particular, sharp upper bounds on *H*2 (2) were obtained in such earlier works as, for example, in [17,18] for various subclasses of the normalized analytic function class A. It is well-known for the Fekete-Szegö functional *a*3 − *a*2 2 that

$$\left| a\_3 - a\_2^2 \right| = H\_2 \left( 1 \right) \dots$$

Its worth mentioning that, for a parameter *μ* which is real or complex, the generalization the functional *a*3 − *μa*2 2 is given in aspects. In particular, Babalola [19] studied the Hankel determinant *H*3 (1) for some subclasses of A.

In 2017, Güney et al. [20] explored the third Hankel determinant in some subclasses of A connected with the above-defined *k*-Fibonacci numbers. A derivation of the sharp coefficient bound for the third Hankel determinant and the conjecture for the sharp upper bound of the second Hankel determinant is also derived by them, which is employed to solve the related problems to the third Hankel determinant and to present an upper bound for this determinant.

Motivated and inspired by the above-mentioned work and also by the recent works of Güney et al. [20] and Uçar [12], we will now define a new subclass SL(*k*, *q*) of starlike functions associated with the *k*-Fibonacci numbers. We will then find the Hankel determinant *H*3 (1) for the newly-defined functions class SL(*k*, *q*).

**Definition 3.** *Let* P (*β*) (0 *β* < 1) *denote the class of analytic functions p in* U *with*

$$p(0) = 1 \quad \text{and} \quad \Re(p\,(z)\,) > \beta.$$

**Definition 4.** *Let the function p be said to belong to the class k-*P˜ *q* (*z*) *and let k be any positive real number if*

$$p\_k(z) \prec \frac{2\overline{p}\_k(z)}{(1+q)+(1-q)\,\overline{p}\_k(z)},\tag{7}$$

*where p*˜*k* (*z*) *is given by*

$$\vec{p}\_k\ (z) = \frac{1 + \mathcal{T}\_k^2 z^2}{1 - k\mathcal{T}\_k z - \mathcal{T}\_k^2 z^2} \,\mathrm{s}\tag{8}$$

*and* T*k is given in* (4)*.*

**Remark 1.** For *q* = 1, it is easily seen that

$$p\left(z\right) \prec p\_k\left(z\right)\left.$$

**Definition 5.** *Let k be any positive real number. Then the function f be in the functions class* SL(*k*, *q*) *if and only if*

$$\frac{z}{f\left(z\right)}\left(\mathcal{D}\_{q}f\right)z \prec \frac{2\vec{p}\_{k}\left(z\right)}{\left(1+q\right)+\left(1-q\right)\vec{p}\_{k}\left(z\right)}\tag{9}$$

*where p*˜*k* (*z*) *is given in* (8)*.*

**Remark 2.** For *q* = 1, we have

$$\frac{zf'(z)}{f(z)} \prec \bar{p}\_k\left(z\right).$$

We recall that when the *f* belongs to the class A of analytic function then it is invariant (or symmetric) under rotations if and only if the function *fς*(*z*) given by

$$f\_{\xi}(z) = e^{-i\xi} f(ze^{i\xi}) \quad (\emptyset \in \mathbb{R})$$

is also in A. A functional I(*f*) defined for functions *f* is in A is called invariant under rotations in A if *fς* ∈ A and I(*f*) = I(*fς*) for all *ς* ∈ R. It can be easily checked that the functionals |*a*2*a*3 − *a*4|, |*H*2,1| and |*H*3,1| considered for the class SL(*k*, *q*) satisfy the above definitions.

**Lemma 1** (see [21])**.** *Let*

$$p(z) = 1 + c\_1 z + c\_2 z^2 + \dots$$

*be in the class* P *of functions with positive real part in* U*. Then*

$$\left|c\_{k}\right| \stackrel{<}{\equiv} 2\tag{10}$$

*If* |*c*1| = 2, *then*

$$p(z) \cong p\_1(z) \cong \frac{1+xz}{1-xz} \qquad \left(x = \frac{c\_1}{2}\right).$$

*Conversely, if p*(*z*) ∼= *p*1(*z*) *for some* |*x*| = 1, *then c*1 = 2*x and*

$$\left| c\_2 - \frac{c\_1^2}{2} \right| \le 2 - \frac{|c\_1^2|}{2}. \tag{11}$$

**Lemma 2** (see [22])**.** *Let p* ∈ P *with its coefficients ck as in Lemma* 1, *then*

$$\left| c\_3 - 2c\_1 c\_2 + c\_1^3 \right| \le 2. \tag{12}$$

**Lemma 3** (see [23])**.** *Let p* ∈ P *with its coefficients ck as in Lemma* 1, *then*

$$|\mathfrak{c}\_1 \mathfrak{c}\_2 - \mathfrak{c}\_3| \le 2.\tag{13}$$

**Lemma 4** (see [20])**.** *If the function f given in the form* (1) *belongs to class* SL*k* , *then*

$$|a\_n| \le |\mathcal{T}\_k|^{n-1} F\_{k, \mathbf{u}^\star} \tag{14}$$

*where* T*k is given in* (4)*. Equality holds true in* (14) *for the function g given by*

$$\begin{aligned} g\_k(z) &= \frac{z}{1 - k\mathcal{T}\_k z - \mathcal{T}\_k^2 z} \\ &= \sum\_{n=1}^{\infty} \mathcal{T}\_k^{n-1} F\_{k,n} z^n, \end{aligned}$$

*which can be written as follows*:

*gk* (*z*) = *z* + T*kz*2 + *k*2 + 1 (T*kk* + 1) *z*3 + ··· . (15)

#### **2. Main Results**

Here, we investigate the sharp bounds for the second Hankel determinant and the third Hankel determinant. We also find sharp bounds for the Fekete-Szegö functional *a*3 − *λa*2 2 for a real number *λ*. Throughout our discussion, we will assume that *q* ∈ (0, 1).

**Theorem 1.** *Let the function f* ∈ A *given in* (1) *belong to the class* SL(*k*, *q*)*. Then*

$$\left| a\_2 a\_4 - a\_3^2 \right| \le \frac{1}{q^3 \left( q + 1 \right)^2 Q} \left\{ \mathcal{Q} \left( q + 1 \right)^2 + \left( |\mathcal{B}\_q| k^2 + |\mathcal{C}\_q| \right) 16 k^2 \right\} \mathcal{T}\_k^2,\tag{16}$$

*where*

$$\mathcal{Q} = \left( q + q^2 + q^3 \right)\_{\epsilon} \tag{17}$$

$$\mathcal{B}\_{\emptyset} = \frac{1}{64} \left( q + 1 \right)^{4} \left\{ \frac{1}{\left( q + 1 \right)^{2}} \mathcal{Q} \left( q^{2} + 6q - 3 \right) - \frac{1}{4} q^{2} \left( q - 1 \right) \left( 2q - 3 \right) \right\} \tag{18}$$

$$\mathcal{C}\_{q} = \frac{1}{16} \left( q + 1 \right)^{2} \left[ \left( 2q - 1 \right) - \mathcal{Q} \left( 3 + \frac{1}{2} q^{2} (q - 1) \right) \left( q + 1 \right)^{2} \right] \tag{19}$$

*and* T*k is given in* (4)*.*

**Proof.** If *f* ∈ SL(*k*, *q*), then it follows from the definition that

$$\frac{z\left(\mathcal{D}\_q f\right)(z)}{f\left(z\right)} \prec \vec{q}\left(z\right)\,\,\prime$$

where

$$\vec{q}\left(z\right) = \frac{2\vec{p}\_k\left(z\right)}{\left(1+q\right) + \left(1-q\right)\vec{p}\left(z\right)}$$

.

For a given *f* ∈ SL(*k*, *q*), we find for the function *p*(*z*), where

$$p\left(z\right) = 1 + p\_1 z + p\_2 z^2 + \dots + \dots$$

that *z*

$$\frac{z\left(\mathcal{D}\_q f\right)(z)}{f(z)} = p\left(z\right) := 1 + p\_1 z + p\_2 z^2 + \dots + p\_q$$

where

If

*p* ≺ *q*˜(*z*).

$$\text{then there is an analytic function } w \text{ such that}$$

$$|w\left(z\right)| \le |z| \quad \text{in} \quad \mathcal{U}$$

*p*(*z*) ≺ *q*˜(*z*),

and

$$p\left(z\right) = \mathfrak{q}\left(w(z)\right).$$

Therefore, the function *g*(*z*), given by

$$\log\left(z\right) = \frac{1+w\left(z\right)}{1-w\left(z\right)} = 1 + c\_1 z + c\_2 z + \cdots \qquad \left(\forall \, z \in \mathbb{U}\right), \tag{20}$$

is in the class P. It follows that

$$w\left(z\right) = \left(\frac{c\_1}{2}\right)z + \left(c\_2 - \frac{c\_1^2}{2}\right)\frac{z^2}{2} + \dotsb$$

and

$$\begin{split} q\left(w\left(z\right)\right) &= 1 + \frac{1}{4}\left(q+1\right)p\_{k,1}c\_{1}z\\ &+ \left[\frac{1}{4}\left(q+1\right)p\_{k,1}\left(c\_{2}-\frac{c\_{1}^{2}}{2}\right)z^{2} + \frac{c\_{1}^{2}}{16}\left(q+1\right)\left[\left(q-1\right)p\_{k,1}^{2} + 2p\_{k,2}\right]\right]z^{2}\\ &+ \left[\frac{1}{4}\left(q+1\right)p\_{k,1}\left(c\_{3}-c\_{1}c\_{2} + \frac{c\_{1}^{3}}{4}\right) + \frac{1}{8}\left(q+1\right)\left\{\left(q-1\right)p\_{k,1}^{2} + 2p\_{k,2}\right\}c\_{1} \\ &\cdot \left(c\_{2}-\frac{c\_{1}^{2}}{2}\right) + \frac{1}{64}\left(q+1\right)\left\{\left(q-1\right)^{2}p\_{k,1}^{3} \quad + 4p\_{k,2}p\_{k,1}\left(q-1\right) + 4p\_{k,3}\right\}c\_{1}^{3}\right]z^{3} + \cdots \\ &= p\left(z\right). \end{split} \tag{21}$$

From (5), we find the coefficient *p*˜*k*,*n* of the function *q*˜ given by

$$
\bar{p}\_{k,n} = \left( F\_{k,n-1} + F\_{k,n+1} \right) T\_k^n.
$$

This shows the following relevant connection *q*˜ with the sequence of *k*-Fibonacci numbers:

$$\begin{aligned} q(w(z)) &= 1 + \frac{1}{4} \left( q + 1 \right) k \mathcal{T}\_k \mathbf{c} \mathbf{1} z + \left[ \frac{1}{4} \left( q + 1 \right) k \mathcal{T}\_k \left( \mathbf{c}\_2 - \frac{\mathbf{c}\_1^2}{2} \right) + \frac{\mathbf{c}\_1^2}{16} (q + 1) \right. \\ & \left. \left. \left( (q - 1) k^2 + 2 \left( 2 + k^2 \right) \right) \mathcal{T}\_k^2 \right] z^2 + \left[ \frac{1}{4} \left( q + 1 \right) k \mathcal{T}\_k \left( \mathbf{c}\_3 - \mathbf{c}\_1 \mathbf{c}\_2 + \frac{\mathbf{c}\_1^2}{4} \right) \right. \\ & \left. + \frac{1}{8} \left( q + 1 \right) \left\{ (q - 1) k^2 + 2 \left( 2 + k^2 \right) \right\} \mathcal{T}\_k^2 \mathbf{c}\_1 \left( \mathbf{c}\_2 - \frac{\mathbf{c}\_1^2}{2} \right) + \frac{1}{64} \left( q + 1 \right) \end{aligned} \tag{22}$$

$$\left. \left\{ (q - 1)^2 k^2 + 4 \left( 2 + k^2 \right) \left( q - 1 \right) + 4 \left( k^2 + 3 \right) \right\} k \mathcal{T}\_k^3 \mathbf{c}\_1^3 \right] z^3 + \cdots$$

If

$$p\left(z\right) = 1 + p\_1 z + p\_2 z^2 + \dotsb \cdot \frac{1}{z}$$

then, by (21) and (22), we find that

$$p\_1 = \left(\frac{q+1}{2}\right) \frac{k\mathcal{T}\_k c\_1}{2} \tag{23}$$

$$p\_2 = \frac{1}{4} \left( q + 1 \right) \left( k \mathcal{T}\_k \left( c\_2 - \frac{c\_1^2}{2} \right) + \frac{c\_1^4}{4} \right) \left\{ \left( q - 1 \right) k^2 + 2 \left( 2 + k^2 \right) \right\} \mathcal{T}\_k^2 \tag{24}$$

$$\begin{split} p\_3 &= (q+1) \left[ \frac{k\mathcal{T}\_k}{2} \left( c\_3 - c\_1 c\_2 + \frac{c\_1^3}{4} \right) + \left\{ (q-1)k^2 \mathcal{T}\_k^2 + 2 \left( 2 + k^2 \right) \mathcal{T}\_k^2 \right\} \right. \\ &\quad \cdot \left( c\_2 - \frac{c\_1^2}{2} \right) \frac{c\_1}{8} + \frac{c\_1^3}{64} \left\{ (q-1)^2 k^2 + 4 \left( 2 + k^2 \right) (q-1) \right\} \\ &\quad + 4 \left( k^2 + 3 \right) \left\{ k \mathcal{T}\_k^3 \right\}. \end{split} \tag{25}$$

Moreover, we have

$$\frac{z\left(\mathcal{D}\_{\emptyset}f\right)(z)}{f\left(z\right)} = 1 + qa\_2z + q\left\{\left(1+q\right)a\_3 - a\_2^2\right\}z^2$$

$$+ \left\{\mathcal{Q}a\_4 - q\left(2+q\right)a\_2a\_3 + qa\_2^3\right\}z^3 + \dots + 1$$

$$= 1 + p\_1z + p\_2z^2 + \dotsb$$

and

$$\begin{aligned} a\_2 &= \frac{p\_1}{q} \\ a\_3 &= \frac{q p\_2 + p\_1^2}{q^2 (q+1)} , \\ a\_4 &= \frac{q^2 (q+1) \, p\_3 - p\_1^3 (q+1) + (2+q) \, \left(p\_1 p\_2 q + p\_1^3\right)}{q^3 (q+1) \, \mathcal{Q}} \end{aligned}$$

Therefore, we obtain

*a*2*a*4 − *a*2 3 = T 2 *k q*3 (*q* + 1) Q (*q* + 1) *c*1 2 2 % Q*c*2 1 16 (*q* + 1) 2 + Q − *q*2 (*q* + 1) 2 *c*2 − *c*2 1 4 (2 + *k*2) 4 & *k* T *n k Fk*,*n* − (*q* + 1) *c*1 2 2 % Q*c*2 1 16 (*q* + 1) 2 Q − *q*2 (*q* + 1) 2 · *c*2 − *c*2 1 4 (2 + *k*2) 4 & *xk*,*n* + (*q* + 1) *k* 2 2 · ⎧ ⎨ ⎩ *q*2 *q* + 1 2 2 *c*1(*c*1*c*2 − *c*3) + Q 4 *c*2 − *c*2 1 2 2 ⎫ ⎬ ⎭ + Q 8 (*q* + 1) − 1 4 *q*2*k*2 *q* + 1 2 *c*4 1 + % E*qk*3 *c*2 1 2 *c*2 − *c*2 1 2 + B*qk*5*c*4 1 + C*qk*3*c*4 1 & T *n k Fk*,*n* − % E*qk*3 *c*2 1 2 *c*2 − *c*2 1 2 + B*qk*5*c*4 1 + C*qk*3*c*4 1 & *xk*,*n* + B*qk*4*c*4 1 + C*qk*2*c*4 1 ,

where

$$\mathcal{E}\_q = \frac{1}{16} \left( q + 1 \right)^2 \left( q - 1 \right) \left\{ \mathcal{Q} - q^2 \left( q + 1 \right)^2 \right\} \dots$$

This can be written as follows:

 *a*2*a*4 − *a*2 3 = T 2 *k q*2 (*q* + 1) 2 Q % Q *q* + 1 2 4 *c*4 1 4 + *q* + 1 2 2 · Q − *q*2 (*q* + 1) 2 *c*2 1 *c*2 − *c*2 1 4 (2 + *k*2) 4 & *k*T *n k Fk*,*n* + *q*2 *q* + 1 2 4 *k*2*c*1(*c*1*c*2 − *c*3) − *q*2 64 (*q* + 1) 4 *k*2*c*4 1 + Q 16 (*q* + 1) 2 *c*4 1 − Q *q* + 1 2 4 *c*4 1 4 *kxk*,*n* + Q*k*2 4 *q* + 1 2 2 *c*2 *c*2 − *c*2 1 2 + 3 8 (*q* + 1) 2 1 6 *k*2 + 2 Q − *q*2 (*q* + 1) 2 *kxk*,*n* − *k*2 4 · *c*2 1 *c*2 − *c*2 1 2 + % E*qk*3 *c*2 1 2 *c*2 − *c*2 1 2 + B*qk*5*c*4 1 + C*qk*3*c*4 1 & T *n k Fk*,*n* − % E*qk*3 *c*2 1 2 *c*2 − *c*2 1 2 + B*qk*5*c*4 1 + C*qk*3*c*4 1 & *xk*,*n* + B*qk*4*c*4 1 + C*qk*2*c*4 1 . (26)

It is known that

$$\forall n \in \mathbb{N}, \ T\_k = \frac{\mathcal{T}\_k^n}{F\_{k,n}} - \mathbf{x}\_{k,n}, \quad \mathbf{x}\_{k,n} = \frac{F\_{k,n-1}}{F\_{k,n}}, \quad \lim\_{n \to \infty} \frac{F\_{k,n-1}}{F\_{k,n}} = \left| \mathcal{T}\_k \right|. \tag{27}$$

Applying (27) together with (11)–(13), we get

 *a*2*a*4 − *a*2 3 T 2 *k q*2 (*q* + 1) 2 Q % Q *q* + 1 2 4 *c*4 1 4 + *q* + 1 2 2 ·{Q − *q*2 (*q* + 1) 2 }*c*2 1 *c*2 − *c*2 1 4 (2 + *k*2) 4 & *k*T *n k Fk*,*n* + *q*2 *q* + 1 2 4 *k*2 |*c*1| |*c*1*c*2 − *c*3| − 1 4 *q*2 *q* + 1 2 *k*2 *c*4 1 + Q 4 *q* + 1 2 2 *c*4 1 − Q *q* + 1 2 4 |*c*4 1| 4 *kxk*,*n* + Q*k*2 4 *q* + 1 2 2 |*c*2| *c*2 − *c*2 1 2 + 3 8 % 2 *q* + 1 2 2 ·{Q − *q*2 (*q* + 1) 2 }*kxk*,*n* − *q* + 1 2 2 *k*2 & |*c*1| 2 · *c*2 − *c*2 1 2 + % E*qk*3 *c*2 1 2 *c*2 − *c*2 1 2 + B*qk*5*c*4 1 + C*qk*3*c*4 1 & T *n k Fk*,*n* − E*qk*3 |*c*2 1| 2 2 − *c*2 1 2 *xk*,*n* − B*qk*5|*c*1| 4*xk*,*n* − C*qk*3|*c*1| 4*xk*,*n* + B*qk*4|*c*4 1| + C*qk*2|*c*4 1|.

From (27), we obtain

$$\left\{ \left( \frac{q+1}{8} \right) \left\{ \mathcal{Q} \left( \frac{q+1}{2} \right) - \mathcal{Q} \left( \frac{q+1}{2} \right)^3 k x\_{k,n} - q^2 k^2 \right\} |c\_1^4| > 0 \right\}$$

and

$$
\left\{ \left( \frac{q+1}{2} \right)^2 \left\{ \frac{2}{3} \left( k^2 + 2 \right) \left\{ \mathcal{Q} - q^2 \left( q+1 \right)^2 \right\} k x\_{k,n} - k^2 \right\} > 0, \right\}
$$

which, for sufficiently large *n*, yields

$$|c\_1| =: y \in \left[0, 2\right].$$

After some computations, we can find that

$$\begin{split} &\max\_{y\in[0,2]} \left\{ \frac{q^2}{8} \left(q+1\right)^4 k^2 y + \left|-\frac{q^2}{8} \left(q+1\right) k^2 + \mathcal{Q} \frac{\left(q+1\right)^2}{16} \right. \\ & \left. - \frac{1}{64} \mathcal{Q} \left(q+1\right)^4 k x\_{\rm{k},n} \right| y^4 + \left|\mathcal{Q} k^2 \frac{\left(q+1\right)^2}{8} \right| \left(2 - \frac{y^2}{2}\right) \\ & \quad + \left|\frac{3}{8} \left\{ \left(\frac{q+1}{2}\right)^2 \left(\frac{2}{3} (k^2+2) \{\mathcal{Q} - q^2 \left(q+1\right)^2\} k x\_{\rm{k},n} - k^2\right) \right\} y^2 \left(2 - \frac{y^2}{2}\right) \\ & \quad - |\mathcal{E}\_q| k^3 \frac{y^2}{2} \left(2 - \frac{y^2}{2}\right) x\_{\rm{k},n} - |\mathcal{B}\_q| k^5 y^4 x\_{\rm{k},n} - |\mathcal{C}\_q| k^3 y^4 x\_{\rm{k},n} + |\mathcal{B}\_q| k^4 y^4 + |\mathcal{C}\_q| k^2 y^4 \left| \begin{matrix} -k + \frac{1}{2} & k - \frac{1}{2} \\ -k & - \frac{1}{2} & k - \frac{1}{2} \end{matrix} \right. \\ & \left. + 4\mathcal{Q} \left(\frac{q+1}{2}\right)^2 \left\{ 1 - k x\_{\rm{k},n} \right\} + \left(16 |\mathcal{B}\_q| k^4 + 16 |\mathcal{C}\_q| k^2 \right) \left\{ 1 - k x\_{\rm{k},n} \right\} \right. \end{split}$$

As a result of the following limit formula:

$$\begin{split} \lim\_{n \to \infty} \left[ \left| \left( \frac{q+1}{2} \right)^2 \frac{c\_1^2}{4} \left\{ \left( \frac{q+1}{2} \right)^2 \mathcal{Q} c\_1^2 + \left\{ \mathcal{Q} - q^2 \left( q+1 \right)^2 \right\} \left( c\_2 - \frac{c\_1^2}{4} \right) (2 + k^2) \right\} \right| \right] \\ &+ \left| \mathcal{E}\_\eta k^3 \frac{c\_1^2}{2} \left( c\_2 - \frac{c\_1^2}{2} \right) + \mathcal{B}\_\eta k^5 c\_1^4 + \mathcal{C}\_\eta k^3 c\_1^4 \right| \right| \frac{|\mathcal{T}\_k^n|}{F\_{k,\text{tr}}} = 0, \end{split}$$

and by using (27), we get

$$\begin{split} \lim\_{n\to\infty} \left[ \max\_{y\in\left[0,2\right]} \left\{ q^2 \left(\frac{q+1}{2}\right)^4 2k^2 y + \left| -\frac{1}{8} q^2 \left(q+1\right) k^2 + \frac{\mathcal{Q}}{16} \left(q+1\right)^2 \right. \right. \\ \left. - \frac{1}{64} \mathcal{Q} \left(q+1\right)^4 k x\_{k,n} \right| y^4 + \mathcal{Q} k^2 \frac{\left(q+1\right)^2}{8} \left(2 - \frac{y^2}{2}\right)^2 + \frac{3}{32} \left(q+1\right)^2 \right. \\ \left. \cdot \left\{ \frac{2}{3} (k^2 + 2) \{ \mathcal{Q} - q^2 \left(q+1\right)^2 \} k x\_{k,n} - k^2 \right\} y^2 \left(2 - \frac{y^2}{2}\right) - |A| k^3 \frac{y^2}{2} \right. \\ \left. \cdot \left( 2 - \frac{y^2}{2} \right) x\_{k,n} - |\mathcal{B}\_q| k^5 y^4 x\_{k,n} - |\mathcal{C}\_{\mathcal{Q}}| k^3 y^4 x\_{k,n} + |\mathcal{B}\_q| k^4 y^4 + |\mathcal{C}\_{\mathcal{Q}}| k^2 y^4 \right) \\ = \mathcal{Q} \left(q + 1\right)^2 \mathcal{T}\_k^2 + \left( |\mathcal{B}\_q| k^2 + |\mathcal{C}\_{\mathcal{Q}}| \right) 16 k^2 \mathcal{T}\_k^2 \end{split}$$

We thus find that

$$\left| a\_2 a\_4 - a\_3^2 \right| \le \frac{\mathcal{T}\_k^2}{q^2 \left( q + 1 \right)^2 \mathcal{Q}} \left\{ \mathcal{Q} \left( q + 1 \right)^2 + \left( |\mathcal{B}\_\eta| k^2 + |\mathcal{C}\_\eta| \right) 16k^2 \right\} \mathcal{T}\_k^2 \dots$$

If, in (20), we set

$$g\_z(z) = \frac{1+z}{1-z} = 1 + 2z + 2z^2 + \dotsb \dotsb$$

then, by putting *c*1 = *c*2 = *c*3 = 2 in (26), we obtain

$$\left| a\_2 a\_4 - a\_3^2 \right| = \frac{\mathcal{T}\_k^2}{q^2 \left( q + 1 \right)^2 \mathcal{Q}} \left\{ \mathcal{Q} \left( q + 1 \right)^2 + \left( |\mathcal{B}\_q| k^2 + |\mathcal{C}\_q| \right) 16k^2 \right\} \mathcal{T}\_k^2 \right\}$$

This completes the proof of Theorem 1.

**Remark 3.** In the next result, for simplicity, we take the values of S*q*, L*q* and M*q* as given by

$$\mathcal{S}\_q = q^3 (1+q) \mathcal{Q}\_\gamma$$

$$\begin{aligned} \mathcal{L}\_{\emptyset} &= \left\{ q \left( 1 + q \right) - q \left( 2 + q \right) + \mathcal{Q} \right\} \left( \frac{q + 1}{2} \right)^3 \left( \frac{k^3 c\_1^3}{8} \right) - q^3 \left( \frac{q + 1}{2} \right)^2 k^3 c\_1^3 \\ &+ \frac{c\_1^3 k}{16} \left[ \left\{ q \mathcal{Q} - q^2 \left( 2 + q \right) \right\} \left( \frac{q + 1}{2} \right)^2 \left\{ \left( q - 1 \right) k^2 + 2 \left( 2 + k^2 \right) \right\} \right] \\ &- \frac{c\_1^3}{32} q^3 \left( \frac{q + 1}{2} \right)^2 \left\{ \left( q - 1 \right) k^3 - \left( q - 1 \right) \left( 2 + k^2 \right) k \right\} \end{aligned}$$

and

$$\mathcal{M}\_q = \left[ \left\{ q\mathcal{Q} - q^2 \left( 2 + q \right) \right\} \left( \frac{q+1}{2} \right)^2 - q^3 \left( \frac{q+1}{2} \right)^3 (q-1)k^2 \right] \frac{c\_1}{2} \left( c\_2 - \frac{c\_1^2}{2} \right) \dots$$

**Theorem 2.** *Let the function f* ∈ A *given in (1) belong to the class* SL(*k*, *q*)*. Then*

$$|a\_2 a\_3 - a\_4| = \frac{2}{\mathcal{S}\_q} \left( \frac{3}{4} k q^3 \left( q + 1 \right)^2 |\mathcal{T}\_k^3| + \frac{1}{2} \left| \left\{ \mathcal{M}\_q + \left( 1 + k^2 \right) \mathcal{L}\_q \right\} \right| k \mathbf{x}\_{k, \mathbf{u}} + \frac{1}{2} k \left| \mathcal{L}\_q \right| \right). \tag{28}$$

**Proof.** Let *f* ∈ SL(*k*, *q*) and let *p* ∈ P be given in (2). Then, from (23)–(25) and

$$\begin{aligned} \frac{z \mathcal{D}\_q f(z)}{f(z)} &= 1 + qa\_2 z + \left\{ \left( q + q^2 \right) a\_3 - q a\_2^2 \right\} z^2 \\ &+ \left\{ \mathcal{Q} a\_4 - \left( 2q + q^2 \right) a\_2 a\_3 + q a\_2^3 \right\} z^3 + \dots + \mathcal{Q} \\ &= 1 + p\_1 z + p\_2 z^2 + \dotsb \end{aligned}$$

we have

$$\begin{aligned} a\_2 a\_3 - a\_4 &= \frac{1}{\mathcal{S}\_q} \left[ \left\{ q \mathcal{Q} - q^2 \left( 2 + q \right) \right\} p\_1 p\_2 \\ &+ \left\{ q \left( 1 + q \right) - q \left( 2 + q \right) + \mathcal{Q} \right\} p\_1^3 - q^3 \left( 1 + q \right) p\_3 \right], \end{aligned}$$

which, together with (27), yields

|*a*2*a*3 − *a*4| = 2 S*q q*3 4 (*q* + 1) 2 6 1 4 *c*2 − *c*3 1 2 *c*1*k*2 + 1 2 (*c*1*c*2 − *c*3) − 3*k*2 + 4 4 *c*1*c*2 *k* T *k fk*,*n* + 1 8 *q*3 (*q* + 1) 2 *c*3 − 2*c*1*c*2 + *c*3 1 *kxk*,*n* + *q*3 8 (*q* + 1) 2 % 1 2 4 − *k*2 *c*2 − *c*2 1 2 + 3*k*2 + 2 *c*2 & *kc*1*xk*,*n* + *q*3 16 (*q* + 1) 2 % 4 − *k*2 *c*2 − *c*2 1 2 − 3*k*2*c*2 & *c*1 − *q*3 32 (*q* + 1) 3 · (*q* − 1) *c*2 − *c*2 1 2 *c*1*k*2 + 1 2 M*q* + 1 + *k*2 L*q k*T*k fk*,*n* −1 2 M*q* + 1 + *k*2 L*q kxk*,*n* + 1 2 *k*L*q* .

In addition, by using (27), we have

$$\frac{q^3}{4}\left(q+1\right)^2\left(4-k^2\right)k\mathbf{x}\_{k,n}-\left(4-k^2\right)<0\qquad(0$$

and

$$\frac{q^3}{4} \left(q+1\right)^2 \left(3k+2\right)k\mathbf{x}\_{k,n} - 3k^2 > 0$$

for 0 < *k* 1 and sufficiently large *n*. Therefore, we have got a function of the variable |*c*1| =: *y* ∈ [0, 2] and, after some computations, we can find that

$$\begin{split} &\max\_{y\in\{0,2\}} \left\{ \frac{q^3}{4} (q+1)^2 \left\{ k \mathbf{x}\_{k,\mathfrak{u}} + \frac{1}{2} \left( k \left( 4 - k^2 \right) \mathbf{x}\_{k,\mathfrak{u}} - \left( 4 - k^2 \right) \right) y \right\} - \frac{q^3}{32} (q+1)^2 \\ &\quad \cdot \left( k \left( 4 - k^2 \right) \mathbf{x}\_{k,\mathfrak{u}} - \left( 4 - k^2 \right) \right) y^3 + \left| \frac{1}{2} \left\{ \mathcal{M}\_{\mathfrak{q}} + \left( 1 + k^2 \right) \mathcal{L}\_{\mathfrak{q}} \right\} \right| k \mathbf{x}\_{k,\mathfrak{u}} + \left| \frac{1}{2} k \mathcal{L}\_{\mathfrak{q}} \right| \right) \\ &= \frac{q^3}{4} (q+1)^2 \left( 3k^2 + 3k \right) \mathbf{x}\_{k,\mathfrak{u}} - 3k^2 + \left| \frac{1}{2} \left\{ \mathcal{M}\_{\mathfrak{q}} + \left( 1 + k^2 \right) \mathcal{L}\_{\mathfrak{q}} \right\} \right| k \mathbf{x}\_{k,\mathfrak{u}} + \left| \frac{1}{2} k \mathcal{L}\_{\mathfrak{q}} \right|. \end{split}$$

As a result of the following limit relation:

$$\begin{aligned} \lim\_{n \to \infty} \left[ \frac{q^3}{16} \left( q + 1 \right)^2 \left( c\_2 - \frac{c\_1^3}{2} \right) c\_1 k^2 + \frac{q^3}{8} \left( q + 1 \right)^2 \left( c\_1 c\_2 - c\_3 \right) - \frac{q^3}{16} \left( q + 1 \right)^2 \right] \\ \quad \cdot \left( 3k^2 + 4 \right) c\_1 c\_2 + \frac{1}{2} \left\{ \mathcal{M}\_q + \left( 1 + k^2 \right) \mathcal{L}\_q \right\} \bigg| \frac{k \mathcal{T}\_k}{f\_{k,n}} = 0 \end{aligned}$$

and, by means of (27), we have

lim*n*→∞ 6 max *y*∈[0,2] *q*3 4 (*q* + 1) 2 *kxk*,*n* + 1 2 *k* 4 − *k*2 *xk*,*n* − 4 − *k*2 *y* − *q*3 32 (*q* + 1) 2 · *k* 4 − *k*2 *xk*,*n* − 4 − *k*2 *y*3 + *q*3 4 (*q* + 1) 2 3*k*2 + 2*k xk*,*n* − 3*k*2 + 1 2 M*q* + 1 + *k*2 L*q kxk*,*n* + 1 2 *k*L*q* = *q*3 *q* + 1 2 2 3*k*2 + 3*k* T*k* − 3*k*2 + 1 2 M*q* + 1 + *k*2 L*q kxk*,*n* + 1 2 *k* L*q* = *q*3 *q* + 1 2 2 −3*k k*2 + 1 T*k* + *k* + 1 2 M*q* + 1 + *k*2 L*q kxk*,*n* + 1 2 *k* L*q* = *q*3 *q* + 1 2 2 −3*k*T 3 *k* ) + 1 2 M*q* + 1 + *k*2 L*q kxk*,*n* + 1 2 *k* L*q* = 3*q*3 *q* + 1 2 2 *k*|T 3 *k* | + 1 2 M*q* + 1 + *k*2 L*q kxk*,*n* + 1 2 *k* L*q*

If, in the formula (20), we set

$$\text{g } (z) = \frac{1+z}{1-z} = 1 + 2z + 2z^2 + \dotsb \text{ : } $$

then, by putting *c*1 = *c*2 = *c*3 = 2 in (26), we obtain

$$|a\_2 a\_3 - a\_4| = \frac{k}{2} \left\{ \frac{3q^3}{\mathcal{S}\_q} \left( q + 1 \right)^2 |\mathcal{T}\_k^3| + 1 \left| \left\{ \mathcal{M}\_q + \left( 1 + k^2 \right) \mathcal{L}\_q \right\} \right| \ge\_{k,n} + |\mathcal{L}\_q| \right\}$$

This completes the proof of Theorem 2.

**Theorem 3.** *Let the function f* ∈ A *given in* (1) *belong to the class* SL(*k*, *q*)*. Then*

$$\left| a\_3 - \lambda a\_2^2 \right| \le \frac{\mathcal{T}\_k^2}{q^2(q+1)} \left[ \left\{ \left| \mathcal{G} \left( \frac{1+q}{2} \right)^2 \right| + \left| \frac{q}{4} \left( q^2 - 1 \right) + q \left( \frac{q+1}{2} \right) \right| \right\} k^2 + q \left( q+1 \right) \right] \tag{30}$$

**Proof.** Let *f* ∈ SL(*k*, *q*) and let *p* ∈ P given in (2). Then, from (23)–(25) and

$$\begin{aligned} \frac{z \mathcal{D}\_q f(z)}{f'(z)} &= 1 + qa\_2 z + \left\{ \left( q + q^2 \right) a\_3 - q a\_2^2 \right\} z^2 \\ &+ \left\{ \mathcal{Q} a\_4 - \left( 2q + q^2 \right) a\_2 a\_3 + q a\_2^3 \right\} z^3 + \dotsb \\ &= 1 + p\_1 z + p\_2 z^2 + \dotsb \end{aligned}$$

we have

$$\left| a\_3 - \lambda a\_2^2 \right| = \frac{1}{q^2 \left( 1 + q \right)} \left| \left[ 1 + \left| \lambda^2 \right| \left( 1 + q \right) \right] p\_1^2 + q p\_2 \right|.$$

Therefore, we obtain

$$\begin{split} \left| a\_3 - \lambda a\_2^2 \right| &= \frac{1}{q^2 \left( 1 + q \right)} \left| (1 - \lambda^2 \left( 1 + q \right)) \left[ \left( \frac{1 + q}{2} \right)^2 \left( \frac{\kappa c\_1 \mathcal{T}}{2} \right)^2 \right] \right| \\ &+ q \left\{ \frac{q + 1}{4} \kappa \mathcal{T}\_k (c\_2 - \frac{c\_1^2}{2}) c\_1^2 (\frac{q + 1}{16}) \right\} \left\{ (q - 1) \kappa^2 + 2 (2 + k^2) \mathcal{T}\_k^2 \right\} \Bigg| \\ &= \frac{\mathcal{T}\_k}{q^2 \left( 1 + q \right)} \left| (1 + \left| \lambda^2 \right| (1 + q) \left( \frac{1 + q}{2} \right) 2 \frac{k^2 c\_1^2}{4} \mathcal{T}\_k + q \left[ \frac{q + 1}{4} \kappa \left( c\_2 - \frac{c\_1^2}{2} \right) \right] \right| \\ &+ \frac{q (q + 1) c\_1^2}{16} (q - 1) k^2 \mathcal{T}\_k + 2 (2 + k^2) \mathcal{T}\_k^2 \Bigg| \Bigg). \end{split}$$

Thus, by applying (27), we have

$$\begin{aligned} \left| a\_3 - \lambda a\_2^2 \right| &= \frac{\mathcal{T}\_k}{q^2 \left( 1 + q \right)} \left| \mathcal{G} \left( \frac{1 + q}{2} \right)^2 \frac{k^2 c\_1^2}{4} \mathcal{T}\_k + q \frac{(q + 1)}{4} \frac{c\_1^2}{4} \right| \\ &\quad \cdot \left[ (q - 1)k^2 + 2(2 + k^2) \frac{\mathcal{T}\_k^n}{f\_{k, \text{tr}}} \right] - \left( \mathcal{G} \left( \frac{1 + q}{2} \right)^2 \frac{k^2 c\_1^2}{4} + q \frac{(q + 1)}{4} \frac{c\_1^2}{4} \right), \\ &\quad \cdot \left[ (q - 1)k^2 + 2(2 + k^2) \right] \mathbf{x}\_{k, \text{tr}} + q \left( \frac{q + 1}{4} \right) k \left( c\_2 - \frac{c\_1^2}{2} \right) \Big| , \end{aligned}$$

where

$$\mathcal{G} = 1 + \left| \lambda^2 \right| \left( 1 + q \right).$$

Now, by applying the triangle inequality in (10)–(13), we have

 T*k q*2 (1 + *q*) G 1 + *q* 2 2 *k*2*c*2 1 4 + *q q* + 1 4 *c*2 1 4 (*q* − 1)*k*2 + 2(2 + *k*2) · T *n k fk*,*n* − 6 G 1 + *q* 2 2 *k*2 4 |*c*1| 2 *xk*,*n* + *q q*2 − 1 4 *k*2 |*c*1| 2 *xk*,*n* + *q q* + 1 4 · *c*2 1 2 (2 + *k*2)*xk*,*n* + *q q* + 1 4 *k c*2 − *c*2 1 2 7 = T*k q*2 (1 + *q*) G 16 (1 + *q*) 2 *k*2*c*2 1 + *q* 16 (*q* + 1) *c*2 1 (*q* − 1)*k*2 + 2(2 + *k*2) · T *n k fk*,*n* − 6 G 1 + *q* 2 2 *k*2 4 |*c*1| 2 *xk*,*n* + *q q* + 1 4 (*q* − 1)*k*2 |*c*1| 2 *xk*,*n* + *q* (*q* + 1) 4 *c*2 1 2 (2 + *k*2)*xk*,*n* + *q q* + 1 4 *k c*2 − *c*2 1 2 7 ,

which, after some computations, yields

$$\begin{split} &\max\_{y\in[0,2]} \left| \mathcal{G} \left( \frac{1+q}{2} \right)^2 \frac{k^2}{4} \right| y^2 \mathbf{x}\_{k,n} + \left| q \left( \frac{q+1}{4} \right) (q-1) k^2 \right| \mathbf{x}\_{k,n} + \left| q \left( \frac{q+1}{4} \right) (2+k^2) \right| \mathbf{x}\_{k,n} \\ & \qquad = \left[ \left| \mathcal{G} \left( \frac{1+q}{2} \right)^2 \right| + \left| q \left( \frac{q^2-1}{4} \right) \right| + \left| q \left( \frac{q+1}{2} \right) \right| \right] k^2 \mathbf{x}\_{k,n} + q(q+1) \mathbf{x}\_{k,n} \end{split}$$

in which we have set *y* = 2. As a result of the following limit formula:

$$\lim\_{n \to \infty} \left| \mathcal{G} \left( \frac{1+q}{2} \right)^2 \frac{k^2 c\_1^2}{4} + q \left( \frac{q+1}{2} \right) \frac{c\_1^2}{4} \left\{ (q-1)k^2 + 2(2+k^2) \right\} \right| \frac{T\_k^n}{f\_{\mathbf{x}\_{k,n}}} = 0, 1$$

which, by applying (27), yields

$$\begin{aligned} \left| a\_3 - \lambda a\_2^2 \right| &\le \frac{\mathcal{T}\_k^2}{q^2(q+1)} \left[ \left\{ \frac{1}{4} \left| \mathcal{G} \left( 1+q \right)^2 \right| + \left| \frac{q}{4} \left( q^2 - 1 \right) + \frac{q}{2} \left( q + 1 \right) \right| \right\} k^2 \right] \\ &+ q \left( q + 1 \right) \right]. \end{aligned}$$

This completes the proof of Theorem 3.

**Theorem 4.** *Let the function f* ∈ A *given in* (1) *belong to the class* SL(*k*, *q*)*. Then*

$$\begin{split} |H\_3(1)| &\leq \frac{\mathcal{T}\_k^6}{q^4(1+q)^3\mathcal{Q}} [(2q+(q+1)k^2)\{16\left|\mathcal{B}\_q\right|k^4+16\left|\mathcal{C}\_q\right|k^2\} \\ &+ \frac{\mathcal{T}\_k^3 2(2+q)k^3+(5q+7)k}{2q^2(1+q)\mathcal{Q}} \left| \{\mathcal{M}\_\emptyset+(1+k^2)\mathcal{L}\_\emptyset\} k\mathbf{x}\_{k,\mathfrak{n}}+k\mathcal{L}\_\emptyset \right|. \end{split}$$

**Proof.** Let *f* ∈ SL(*k*, *q*). Then as we know that

$$H\_3(1) = \begin{vmatrix} a\_1 & a\_2 & a\_3 \\ a\_2 & a\_3 & a\_4 \\ a\_3 & a\_4 & a\_5 \end{vmatrix} = a\_3(a\_2a\_4 - a\_3^2) - a\_4(a\_4 - a\_2a\_3) + a\_5(a\_3 - a\_2^2),$$

where *a*1 = 1 so, we have

$$\begin{array}{c} |H\_3(1)| \le |a\_3| \left| a\_2 a\_4 - a\_3^2 \right| + |a\_4| \left| a\_4 - a\_2 a\_3 \right| + |a\_5| \left| a\_3 - a\_2^2 \right| \end{array} \tag{31}$$

Thus, by using Lemma 4, Theorems 1–3, as well as the formula (31), we find that

$$|H\_3(1)| \le 2(\mathbb{P}\_{\mathfrak{q}}k^6 + 4\mathbb{P}\_{\mathfrak{q}}k^4 + 4\mathbb{Y}\_{\mathfrak{q}}k^2 + \Gamma\_{\mathfrak{q}})\mathcal{T}\_k^6. \tag{32}$$

{M*q* + (1 + *k*2)*Lq*}*kxk*,*n* + *kLq*

 ,

where

*q* = κ*q*,*λ* 2*q*2Q % 1 + *χq*(2 + *q*) Q*q*2 − (*q* + 3)(*q* + 1)2 4*q*2 + *q* + 1 *q*3 + (*q* + 1) 4 16*q*3 & Ψ*q* = 3(2 + *q*) *q*3(1 + *q*)2Q + κ*q*,*λ* 2*q*2(Q + 1) 4 + (5*q* + 7)*χq* 2*q*2Q − (*q* + 3)(*q* + 1) 2*q* + 4 *q*2 + (*q* + 1) 2*q*(Q + 1) % 1 + *χq*(2 + *q*) *q*2Q − (*q* + 3)(*q* + 1)2 4*q*2 + *q* + 1 *q*3 + *q q* + 1 2*q* 4 & Υ*q* = 1 *q*2 + 1 (Q + 1) 4 + *χq*(5*q* + 7) 2*q*2Q − *q* + 1 2*q* (*q* + 1)(*q* + 3) 2*q* + 4 *q*2 + 3(5*q* + 7) 2*q*2Q + κ*q*,*λ* 2*q*2(Q + 1) 2 + 4 *q*(1 + *q* Γ*q* = 1 *q*4(1 + *q*)3Q 2*q* + (*q* + 1)*k*2 16 B*q k*4 + 16 C*q k*2 2(2 + *q*)*k*3 + (5*q* + 7)*k*

and

$$
\chi\_q = q^2 + q + 2.
$$

This completes the proof of Theorem 4.

+

2*q*2(1 + *q*)Q

#### **3. Conclusions**

A new subclass of analytic functions associated with *k*-Fibonacci numbers has been introduced by means of quantum (or *q*-) calculus. Upper bound of the third Hankel determinant has been derived for this functions class. We have stated and proved our main results as Theorems 1–4 in this article.

Further developments based upon the the *q*-calculus can be motivated by several recent works which are reported in (for example) [24,25], which dealt essentially with the second and the third Hankel determinants, as well as [26–29], which studied many different aspects of the Fekete-Szegö problem.

**Author Contributions:** Conceptualization, H.M.S. and Q.Z.A.; methodology, Q.Z.A.; formal analysis, H.M.S. and M.D.; Investigation, M.S.; resources, X.X.; data curation, S.K.; writing—review and editing, N.K.; visualization, N.K.; funding acquisition, M.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by UKM Grant: FRGS/1/2019/STG06/UKM/01/1.

**Acknowledgments:** The work here is supported by UKM Grant: FRGS/1/2019/STG06/UKM/01/1.

**Conflicts of Interest:** The authors declare that they have no conflicts of interest.

#### **References**

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Difference of Some Positive Linear Approximation Operators for Higher-Order Derivatives**

**Vijay Gupta 1,†, Ana Maria Acu 2,† and Hari Mohan Srivastava 3,4,5,\*,†**

Received: 21 April 2020; Accepted: 25 May 2020; Published: 2 June 2020

**Abstract:** In the present paper, we deal with some general estimates for the difference of operators which are associated with different fundamental functions. In order to exemplify the theoretical results presented in (for example) Theorem 2, we provide the estimates of the differences between some of the most representative operators used in Approximation Theory in especially the difference between the Baskakov and the Szász–Mirakyan operators, the difference between the Baskakov and the Szász–Mirakyan–Baskakov operators, the difference of two genuine-Durrmeyer type operators, and the difference of the Durrmeyer operators and the Lupa¸s–Durrmeyer operators. By means of illustrative numerical examples, we show that, for particular cases, our result improves the estimates obtained by using the classical result of Shisha and Mond. We also provide the symmetry aspects of some of these approximations operators which we have studied in this paper.

**Keywords:** approximation operators; differences of operators; Szász–Mirakyan–Baskakov operators; Durrmeyer type operators; Bernstein polynomials; modulus of continuity

#### **1. Introduction, Definitions and Preliminary Results**

Approximation by positive linear operators is a classical and important topic of research in Approximation Theory and Computer-Aided Geometric Design (CAGD). The basis of the familiar Bernstein operators is an important tool in Computer-Aided Geometric Design. This basis is used in order to construct Bézier curves, which have applications for designing curves for the cars industry and problems involving animations. In addition, the Bézier curves are used in order to control the velocity over time. A class of symmetric Beta-type distributions involving the symmetric Bernstein-type basis function was introduced and studied in [1]. In recent years, the quantum (or the *q*-) calculus and its variation, the so-called post-quantum or the (*p*, *q*)-calculus, which have many applications in quantum physics, attracted the attention of many researchers. For example, some variations of positive linear operators by using the (*p*, *q*)-calculus instead of their known forms involving the traditional *q*-calculus were, in fact, published recently in *Symmetry* itself (see [2]). In this connection, the readers are referred also to a subsequent survey-cum-expository review article by Srivastava [3] in which the above-mentioned variation aspect of the (*p*, *q*)-calculus was exposed. Several other applications of the positive linear operators in learning theory can also be found in the literature. For more details about

this topic, the reader is referred to the applications of the Bernstein operators and the iterated Boolean sums of operators (see [4]) and the applications of the Durrmeyer operators (see [5]).

The attention of many researchers in the study of the differences of positive linear operators began with the question raised by Lupa¸s in regard with the possibility to give an estimate for the following commutator:

$$[B\_n, \overline{\mathbb{B}}\_n] := B\_n \diamond \overline{\mathbb{B}}\_n - \overline{\mathbb{B}}\_n \diamond B\_{n\nu}$$

where *Bn* are the Bernstein operators and B*n* are the Beta operators (see, for details, [6]).

In [7], an algebraic structure of positive linear operators, which map *C*[0, 1] into itself, was considered in order to give an inequality for the commutators of certain positive linear operators. In several sequels to this study, Gonska et al. (see, for example, [8–10]) considered an algebraic structure (*S*, +, ◦, 0, *I*) which satisfies each of the following conditions:

The set

$$PLO = \{ L : \mathbb{C}[0, 1] \to \mathbb{C}[0, 1] \quad \text{and} \quad L \text{ is linear and positive} \} \text{-} $$

which is equipped with the canonical operations of addition and operator composition, is an algebraic structure defined above. The commutator given by

$$[A,B] := AB - BA \qquad (A, B \in PLO)$$

was studied from a quantitative point of view in [7].

A solution of the Lupa¸s problem was given by Gonska et al. [7] by using the Taylor expansion. The estimates for the differences of two positive linear operators, which have the same moments up to a certain order, were derived in [8–10]. In [11], the differences of certain positive linear operators, which have the same fundamental functions, were studied. These studies of the positive linear operators, which are defined on unbounded interval, become an interesting area of research in Approximation Theory (see [12–15]). Estimates for the differences of these operators in terms of weighted modulus of smoothness were obtained by Aral et al. [16]. The Bernstein polynomials are, by all means, the most investigated polynomials in Approximation Theory and were introduced by Bernstein in order to prove the Weierstrass Theorem. Various new generalizations of these operators were considered in, for example, [17,18]. In [19], estimates of the differences of the Bernstein operators and their derivatives were obtained. Recently, some interesting results on this topic were published in [20–25]. In the present paper, our approach involves positive linear operators which have substantially different fundamental functions. In fact, the results presented in this paper extend the earlier studies in [11] for more general classes of positive linear operators.

We denote by *E*(*I*) the space of real-valued continuous functions defined on an interval *I* ⊆ R, which contains the polynomials. Let

$$||f|| = \sup\left\{|f(x)|: x \in I\right\}$$

and

$$E\_B(I) := \{ f \in E(I) \quad \text{and} \quad ||f|| < \infty \}.$$

Let *ej*(*t*) := *tj* (*j* = 0, 1, 2, ··· . We consider the linear positive functional *F* : *E*(*I*) → *R* preserving constant function, namely, *F*(*e*0) = 1. We also put

$$\mu\_r^F = F\left( (\varepsilon\_1 - \boldsymbol{\phi}^F \boldsymbol{\varepsilon}\_0)^r \right) := \sum\_{i=0}^r \binom{r}{i} (-1)^i \, ^i F(\boldsymbol{\varepsilon}\_{r-i}) [\boldsymbol{\phi}^F]^i \quad (r \in \mathbb{N}),$$

where *φF* := *F*(*e*1). For the functional *F*, the following basic result was obtained in [11].

**Lemma 1** (see [11])**.** *Let f* ∈ *E*(*I*) *with f* (4) ∈ *EB*(*I*)*. Then*

$$\left| F(f) - f(\phi^F) - \frac{\mu\_2^F}{2!} f^{(2)}(\phi^F) - \frac{\mu\_3^F}{3!} f^{(3)}(\phi^F) \right| \le \frac{\mu\_4^F}{4!} ||f^{(4)}||.$$

Let us now consider the fundamental functions *pm*,*k*, *bm*,*k* ≥ 0, *k* ∈ *K*, and *pm*,*k*, *bm*,*k* ∈ *C*(*I*) such that

$$\sum\_{k \in K} p\_{m,k}(\mathbf{x}) = \sum\_{k \in K} b\_{m,k}(\mathbf{x}) = \varepsilon\_{0\prime}$$

where *K* is a set of non-negative integers, that is,

$$K = \mathbb{N}^0 := \mathbb{N} \cup \{0\} .$$

Suppose also that *Fm*,*k*, *Gm*,*k* : *E*(*I*) → *R* are the linear positive functionals such that

$$F\_{m,k}(e\_0) = G\_{m,k}(e\_0) = 1$$

and denote

$$D(I) := \left\{ f \in E(I) \, \middle| \, \sum\_{k \in K} p\_{m,k} F\_{m,k}(f) \in \mathbb{C}(I) \text{ and } \sum\_{k \in K} b\_{m,k} G\_{m,k}(f) \in \mathbb{C}(I) \right\}.$$

Define the positive linear operators *Um*, *Vm* : *D*(*I*) → *C*(*I*) as follows:

$$\mathcal{U}\_{\mathfrak{m}}(f,\mathfrak{x}) := \sum\_{k \in K} p\_{m,k}(\mathfrak{x}) F\_{m,k}(f) \quad \text{and} \quad V\_{\mathfrak{m}}(f,\mathfrak{x}) := \sum\_{k \in K} b\_{m,k}(\mathfrak{x}) G\_{m,k}(f).$$

In [11], the following result concerning the difference of the operators *Um* and *Vm* was proved.

**Theorem 1** (see [11])**.** *Suppose that*

$$p\_{m,k} = b\_{m,k} \quad \text{and} \quad \phi^{F\_{m,k}} = \phi^{G\_{m,k}} \quad k \in \mathcal{K}; \ m \in \mathbb{N}.$$

*Let f* ∈ *D*(*I*) *with f* (*i*) ∈ *EB*(*I*) (*i* = 2, 3, 4)*. Then*

$$|(\mathcal{U}\_m - V\_m)(f, \mathbf{x})| \le ||f^{(2)}||\gamma(\mathbf{x}) + ||f^{(3)}||\beta(\mathbf{x}) + ||f^{(4)}||\boldsymbol{a}(\mathbf{x}) \quad (\mathbf{x} \in I),$$

*where*

$$\begin{aligned} \gamma(\mathbf{x}) &:= \sum\_{k \in K} |\mu\_2^{F\_{m,k}} - \mu\_2^{G\_{m,k}}| p\_{m,k}(\mathbf{x})\_\mathbf{n} \\\\ \circledast \gamma &:= \sum\_{k \in K} F\_{m,k} - \mu\_{m,k}^{G\_{m,k}} \mathbf{n} \quad (\mathbf{x}) \end{aligned}$$

$$\beta(\mathbf{x}) := \sum\_{k \in K} |\mu\_3^{F\_{m,k}} - \mu\_3^{G\_{m,k}}| p\_{m,k}(\mathbf{x})$$

*and*

$$\mathfrak{a}(\mathfrak{x}) := \sum\_{k \in \mathcal{K}} (\mu\_4^{F\_{m,k}} + \mu\_4^{G\_{m,k}}) p\_{m,k}(\mathfrak{x}).$$

In the series of papers [8–10], the results concerning the estimations of the differences of certain positive linear operators were based upon the fact that the positive linear operators have the same moments up to a certain order. In the recent paper [11], the approach involved the positive linear operators which have the same fundamental functions. The main goal of this paper is to extend the above result for the positive linear operators that have different fundamental functions. Furthermore, the condition *φFm*,*k* = *φGm*,*k* of ([11], Theorem 4) is shown to be not necessary in order to obtain an estimate of the differences of the positive linear operators *Vm* and *Um*.

**Theorem 2.** *Let f* ∈ *D*(*I*)*. If f* (*i*) ∈ *EB*(*I*) (*i* = 2, 3, 4), *then*

$$\begin{aligned} \left| (\mathcal{U}\_m - V\_m)(f, \mathbf{x}) \right| &\le A(\mathbf{x}) \| f^{(4)} \| + B(\mathbf{x}) \| f^{(3)} \| + C(\mathbf{x}) \| f^{(2)} \| \\ &+ 2\omega\_1(f, \delta\_1(\mathbf{x})) + 2\omega\_1(f, \delta\_2(\mathbf{x})) \quad (\mathbf{x} \in I), \end{aligned}$$

*where ω*1(*f* , ·) *is the usual modulus of continuity,*

$$\begin{aligned} A(\mathbf{x}) &= \frac{1}{4!} \sum\_{k \in K} (p\_{m,k}(\mathbf{x}) \boldsymbol{\mu}\_4^{\mathrm{F}\_{m,k}} + b\_{m,k}(\mathbf{x}) \boldsymbol{\mu}\_4^{\mathrm{G}\_{m,k}}), \\\\ B(\mathbf{x}) &= \frac{1}{3!} \left| \sum\_{k \in K} p\_{m,k}(\mathbf{x}) \boldsymbol{\mu}\_3^{\mathrm{F}\_{m,k}} - \sum\_{k \in K} b\_{m,k}(\mathbf{x}) \boldsymbol{\mu}\_3^{\mathrm{G}\_{m,k}} \right|, \\\\ C(\mathbf{x}) &= \frac{1}{2!} \left| \sum\_{k \in K} p\_{m,k}(\mathbf{x}) \boldsymbol{\mu}\_2^{\mathrm{F}\_{m,k}} - \sum\_{k \in K} b\_{m,k}(\mathbf{x}) \boldsymbol{\mu}\_2^{\mathrm{G}\_{m,k}} \right|, \\\\ \delta\_1(\mathbf{x}) &= \left( \sum\_{k \in K} p\_{m,k}(\mathbf{x}) \left( \boldsymbol{\Phi}^{\mathrm{F}\_{m,k}} - \mathbf{x} \right)^2 \right)^{1/2} \end{aligned}$$

*and*

$$\delta\_2(\mathbf{x}) = \left(\sum\_{k \in K} b\_{m,k}(\mathbf{x}) \left(\boldsymbol{\phi}^{G\_{m,k}} - \mathbf{x}\right)^2\right)^{1/2} \cdot \mathbf{x}$$

**Proof.** First of all, by using Lemma 1, we get

$$\begin{split} & \left| (\boldsymbol{M}\_{m} - \boldsymbol{V}\_{m}) (\boldsymbol{f}, \boldsymbol{x}) \right| \leq \left| \sum\_{k \in K} p\_{m,k}(\boldsymbol{x}) F\_{m,k}(\boldsymbol{f}) - \sum\_{k \in K} b\_{m,k}(\boldsymbol{x}) G\_{m,k}(\boldsymbol{f}) \right| \\ & \leq \sum\_{k \in K} p\_{m,k}(\boldsymbol{x}) \left| F\_{m,k}(\boldsymbol{f}) - f(\boldsymbol{\Phi}^{\boldsymbol{x}\_{k}}) - \frac{\mu\_{2}^{\boldsymbol{F}\_{\boldsymbol{m},k}}}{2!} f''(\boldsymbol{\Phi}^{\boldsymbol{F}\_{\boldsymbol{m},k}}) - \frac{\mu\_{3}^{\boldsymbol{F}\_{\boldsymbol{m},k}}}{3!} f'''(\boldsymbol{\Phi}^{\boldsymbol{F}\_{\boldsymbol{m},k}}) \right| \\ & + \sum\_{k \in K} b\_{m,k}(\boldsymbol{x}) \left| G\_{m,k}(\boldsymbol{f}) - f(\boldsymbol{\Phi}^{\boldsymbol{G}\_{\boldsymbol{m},k}}) - \frac{\mu\_{2}^{\boldsymbol{G}\_{\boldsymbol{m},k}}}{2!} f'''(\boldsymbol{\Phi}^{\boldsymbol{G}\_{\boldsymbol{m},k}}) - \frac{\mu\_{3}^{\boldsymbol{G}\_{\boldsymbol{m},k}}}{3!} f'''(\boldsymbol{\Phi}^{\boldsymbol{G}\_{\boldsymbol{m},k}}) \right| \\ & + \left| \sum\_{k \in K} p\_{m,k}(\boldsymbol{x}) \frac{\mu\_{2}^{\boldsymbol{F}\_{\boldsymbol{m},k}}}{2!} - \sum\_{k \in K} b\_{m,k}(\boldsymbol{x}) \frac{\mu\_{2}^{\boldsymbol{G}\_{\boldsymbol{m},k}}}{2!} \right| \cdot ||f''|| \end{split}$$

+ ∑ *k*∈*K pm*,*k*(*x*) *μFm*,*k* 3 3! − ∑ *k*∈*K bm*,*k*(*x*) *μGm*,*k* 3 3! · || *f* || + ∑ *k*∈*K pm*,*k*(*x*)| *f*(*φFm*,*k* ) − *f*(*x*)| + ∑ *k*∈*K bm*,*k*(*x*)| *f*(*φGm*,*k* ) − *f*(*x*)| ≤ 1 4! ∑ *k*∈*K pm*,*k*(*x*)*μFm*,*k* 4 + ∑ *k*∈*K bm*,*k*(*x*)*μGm*,*k* 4 *f* (*iv*) + ∑ *k*∈*K pm*,*k*(*x*) *μFm*,*k* 2 2! − ∑ *k*∈*K bm*,*k*(*x*) *μGm*,*k* 2 2! · || *f* || + ∑ *k*∈*K pm*,*k*(*x*) *μFm*,*k* 3 3! − ∑ *k*∈*K bm*,*k*(*x*) *μGm*,*k* 3 3! · || *f* || + ∑ *k*∈*K pm*,*k*(*x*)| *f*(*φFm*,*k* ) − *f*(*x*)| + ∑ *k*∈*K bm*,*k*(*x*)| *f*(*φGm*,*k* ) − *f*(*x*)| = *A*(*x*) *f* (4) + *B*(*x*) *f* (3) + *C*(*x*) *f* (2) + 1 + ∑*k*∈*K pm*,*k*(*x*) *φFm*,*k* − *x* 2 *δ*2 1 (*x*) *ω*1(*f* , *δ*1(*x*)) + 1 + ∑*k*∈*K bm*,*k*(*x*) *φGm*,*k* − *x* 2 *δ*2 2 (*x*) *ω*1(*f* , *δ*2(*x*)) = *A*(*x*) *f* (4) + *B*(*x*) *f* (3) + *C*(*x*) *f* (2) + 2*ω*1(*f* , *δ*1(*x*)) + 2*ω*1(*f* , *δ*2(*x*)).

This completes the proof of Theorem 2.

**Remark 1.** *Let*

$$\nu\_1(\mathfrak{x}) = \left( \mathcal{U}\_m\left( (e\_1 - \mathfrak{x})^2; \mathfrak{x} \right) \right)^{\frac{1}{2}}$$

*and*

$$\nu\_2(\mathbf{x}) = \left(V\_m\left((e\_1 - \mathbf{x})^2; \mathbf{x}\right)\right)^{\frac{1}{2}}.$$

*Then, by using the result of Shisha and Mond [26], we find that*

$$\begin{aligned} |(\mathcal{U}\_{\textit{m}} - V\_{\textit{m}})(f; \mathbf{x})| &\leq |\mathcal{U}\_{\textit{m}}(f; \mathbf{x}) - f(\mathbf{x})| + |V\_{\textit{m}}(f; \mathbf{x}) - f(\mathbf{x})| \\ &\leq 2\omega\_{1}(f, \nu\_{1}(\mathbf{x})) + 2\omega\_{1}(f, \nu\_{2}(\mathbf{x})). \end{aligned}$$

*Since*

$$F\_{m,k}^2(e\_1) \le F\_{m,k}(e\_1^2)$$

*and*

$$G\_{m,k}^2(e\_1) \le G\_{m,k}(e\_1^2),$$

*it follows that*

$$
\delta\_i(\mathfrak{x}) \le \nu\_i(\mathfrak{x}) \qquad (i = 1, 2).
$$

#### **2. Applications of Theorem 2**

As applications of the Theorem 2, in this section, we give estimates of the differences between some of the most used positive linear operators in Approximation Theory. The considered examples involve the Baskakov type operators, the Szász–Mirakyan type operators, and the Durrmeyer type operators. We also show for the Durrmeyer type operators that, in some particular cases, our result improves the estimates obtained by using the classical result of Shisha and Mond [26].

#### *2.1. Difference Between the Baskakov and the Szász–Mirakyan Operators*

The Szász–Mirakyan operators are defined by

$$S\_m(f, \mathbf{x}) = \sum\_{k=0}^{\infty} p\_{m,k}(\mathbf{x}) F\_{m,k}(f),\tag{1}$$

where

$$p\_{m,k}(x) = e^{-mx} \frac{(mx)^k}{k!} \quad \text{and} \quad F\_{m,k}(f) = f\left(\frac{k}{m}\right).$$

**Lemma 2.** *The moments of Sm satisfy the following relation*:

$$S\_{\mathfrak{m}}(e\_{\mathfrak{n}+1}, \mathfrak{x}) = \frac{\mathfrak{x}}{m} S\_{\mathfrak{m}}'(e\_{\mathfrak{n}}, \mathfrak{x}) + \mathfrak{x} S\_{\mathfrak{m}}(e\_{\mathfrak{n}}, \mathfrak{x}) .$$

*In particular*,