**5. Conclusion**

In this chapter, we introduced agent-based modeling of cancer evolution along with methodologies for data fitting and sensitivity analysis. Although there is a long history of theoretical science in the field of cancer research, this approach has been overshadowed by experimental science until recently. However, with a recent explosive increase in cancer genome data, there is now an increasing need to integrate experimental and theoretical science. As an example, this chapter introduced methods for modeling and analyzing the evolutionary processes generating ITH, which is experimentally observed by multiregion sequencing. We also presented exemplifying applications: e.g., agent-based simulation modeling and analysis successfully demonstrated that ITH in colorectal cancer is generated by neutral evolution, which is caused by a high mutation rate and stem cell hierarchy. For cancer genome analyses, new experimental technologies are actively being developed. For example, single-cell sequencing technologies can profile IHT at the ultimate resolution [30] while liquid biopsy technologies, such as the sequencing of circulating tumor DNA, enables us to non-invasively track cancer evolution during treatment [31]. These technologies will unveil more various aspects of cancer evolution when combined with the approach introduced in this chapter. This chapter also exemplified how simulation modeling helps to solve scientific problems raised by new experimental technologies. We hope that this chapter will provides readers with some hints to solve their own problems using simulation modeling.


**6. Conclusions**

This study explores the effect of radar reflectivity and GTS data assimilation from assimilation frequency using WRF-3DVar for rainfall forecasting. Four heavy storm events at the Daqinghe catchment in the Beijing Tianjin Hebei region of northern China are selected to be regenerated by the WRF model. We employed three nested domains, and adopted the GFS data for driving the WRF model. From two aspects of cumulative rainfall and spatial distribution of rainfall, two observational data types (radar reflectivity and GTS data) assisted in investigating how WRF rainfall forecasts were potentially improved in space and time through data assimilation. We designed four data assimilation schemes considering various possible combinations of the two data assimilation frequency types in the three nested domains. We compared the analysis with data assimilation and that without data assimilation, finding that the assimilation results partly fit observations in the case and that WRF-3DAVR with radar reflectivity and GTS data better represents the rainfall forecasts in space and time.

Precipitation simulated by the WRF model is always much lower than observed rainfall, but assimilation systems can increase rainfall. The improved initial conditions in WRF-3DVAR system via radar data assimilation and GTS data achieved better shortterm and convective strong precipitation in the both temporal dimension and spatial dimension. The high assimilation frequency significantly helps to trigger and maintain the convective activities in the 3DVAR framework as well as the storm case applied. Forecasts of events indicate that the temporal rainfall distributions of convective storms can be much better predicted with high assimilation frequency, compared with the 6 h assimilation time interval run. At the same time, employing the high assimilation frequency to the assimilation showed improved skill of precipitation forecasting in WRF-3DVAR on spatial

rainfall distributions. The only exception happened in Event II. In this case, the impact of false rainfall forecasting fields is enlarged because of the inaccurate radar observed data, so that the negative impact was found after assimilation. However, this does not seem to pose a threat to the hourly assimilation frequency. In addition, the data assimilation of outside domain has small impact on output of inside domain in not only temporal but also spatial dimension. In general, the hourly data assimilation frequency together with strict outputs from domain resolutions is closer to actual precipitation. In this study, the assimilation by combining the radar reflectivity and Global Tele-communication System (GTS) data with high assimilation frequency is helpful for further enhancing the temporal and spatial distribution of the short-term precipitation forecast. The results can be used as a reference for areas with similar climatic conditions as well as rainfall characteristics. The methodology is of guiding significance for WRF-3DAVR rainfall forecasting. In this case, in order to explore universally applicable data assimilation guidelines for rainfall forecasting, research should be conducted over more storm events in different study areas.

**Author Contributions:** Conceptualization, Y.L. and J.L.; Methodology, Y.L., J.L. and F.Y.; Software, Y.L. and Y.L.; Validation, Y.L. and W.W.; Formal analysis, Y.L. and J.L.; Investigation, W.W., C.L. and F.Y.; Writing-original draft preparation, Y.L. and J.L.; Writing-review and editing, Y.L., J.L. and C.L.; Visualization, Y.L.; Funding acquisition, J.L. and C.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Major Science and Technology Program for Water Pollution Control and Treatment (2018ZX07110001), the National Natural Science Foundation of China (51822906), the National Key Research and Development Project (2017YFC1502405), and the IWHR Research & Development Support Program (WR0145B732017).

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


**2.2.2 Yeasts in the equine GIT**

Several investigators focused on the effect of yeast, *Saccharomyces cerevisiae,* on intestinal microbiota of horses and on the digestibility of different diets (e.g. Medina et al., 2002; Jouany et al., 2008; 2009). But little is known about the yeasts naturally occurring in the equine gut. Van Uden et al., (1958) studied yeasts in the caecal contents of 252 horses, and these authors revealed presence of yeasts in over half (52.4%) of the investigated animals. With occurrence of 21.8%, *Trichosporon cutaneum* was the most frequently isolated yeast, whereas in porcine intestine this species was found only one time. However, other yeasts: *C. krusei* (8.3%) as well as *C. tropicalis* (4.4%), *C. albicans* (4.4%), *C. parapsilosis* (3.6%), *C. slooffiae*  (2.4%), *C. glabrata* (1.2%) and *S. telluris* (1.2%) detected in the GIT of horses were also commonly recorded in pigs.


*Article* **On Some New Inequalities of Hermite–Hadamard Midpoint and Trapezoid Type for Preinvex Functions in (***p***,** *q***)-Calculus**

**Ifra Bashir Sial 1, Muhammad Aamir Ali 2,\*, Ghulam Murtaza 3, Sotiris K. Ntouyas 4,5, Jarunee Soontharanon 6 and Thanin Sitthiwirattham 7,\***

**Abstract:** In this paper, we establish some new Hermite–Hadamard type inequalities for preinvex functions and left-right estimates of newly established inequalities for (*p*, *q*)-differentiable preinvex functions in the context of (*p*, *q*)-calculus. We also show that the results established in this paper are generalizations of comparable results in the literature of integral inequalities. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role.

**Keywords:** Hermite–Hadamard inequality; (*p*, *q*)-integral; post quantum calculus; convex function

## **1. Introduction**

The Hermite–Hadamard (H-H) inequality, which was independently found by C. Hermite and J. Hadamard, is particularly important in convex function theory (see, [1–3], and also [4], p. 137).

$$\Pi\left(\frac{\pi\_1 + \pi\_2}{2}\right) \le \frac{1}{\pi\_2 - \pi\_1} \int\_{\pi\_1}^{\pi\_2} \Pi(\varkappa) d\varkappa \le \frac{\Pi(\pi\_1) + \Pi(\pi\_2)}{2} \tag{1}$$

where Π is a convex mapping. The aforementioned inequality is true in reverse order for concave maps. Jensen's inequality for convex functions can easily capture this inequality [5]. Several generalizations and extensions to classical convex functions have been proposed in recent years. In [6], the notions about invex function was given that is a significant generalization of convex functions. Weir and Mond introduced the concept of preinvex functions in [7], and it is used in optimization theory in a variety of ways. Prequasi-invex functions are a generalization of the invex functions introduced by Pini in [8]. Following that, the authors looked at some fundamental properties of generalized preinvex functions in [9]. Noor established H-H integral inequalities for preinvex functions in [10–12]. The authors of [13,14] used the ordinary and fractional integrals to calculate the left and right bounds of the H-H inequalities for preinvex functions. More recent results on the integral inequalities for various types of preinvexities can be found in [15–24].

**Citation:** Sial, I.B.; Ali, M.A.; Murtaza, G.; Ntouyas, S.K.; Soontharanon, J.; Sitthiwirattham, T. On Some New Inequalities of Hermite–Hadamard Midpoint and Trapezoid Type for Preinvex Functions in (*p*, *q*)-Calculus. *Symmetry* **2021**, *13*, 1864. https:// doi.org/10.3390/sym13101864

Academic Editor: Nicusor Minculete

Received: 7 September 2021 Accepted: 26 September 2021 Published: 3 October 2021

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

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

On the other hand, beginning with Euler, various efforts in the subject of *q*-analysis have been implemented in order to master the mathematics that underpins quantum computing. The phrase *q*-calculus binds mathematics and physics together. It is employed in subjects including combinatorics, number theory, basic hypergeometric functions, orthogonal polynomials, and others, as well as relativity theory, mechanics, and quantum theory [25,26]. It has numerous applications in quantum information theory [27,28]. Euler is believed to be the creator of this crucial branch of mathematics since he employed the *q*-parameter in Newton's work on infinite series. Jackson [29,30] introduced the concept of *q*-calculus, sometimes known as calculus without limits, for the first time in a proper manner. Al-Salam [31] introduced the concepts of *q*-fractional integral and *q*-Riemann–Liouville fractional integral in 1996. Because study in this subject is gradually increasing, Tariboon and Ntouyas [32] proposed the *π*1*Dq*-difference operator and *qπ*1 -integral. Bermudo et al. [33] published their ideas regarding the *π*2*Dq*-difference operator and *qπ*2 -integral in 2020. By presenting the principles of (*p*, *q*)-calculus, Sadjang [34] broadened the concept of *q*-calculus. Tunç and Göv [35] introduced the (*p*, *q*)-variant of the *π*1*Dq*-difference operator and *qπ*1 -integral. Chu et al. established the concepts of *π*2*Dp*,*q*-derivative and (*p*, *q*) *π*2 -integral in [36], in 2021.

Quantum and post-quantum integrals have been used to study a variety of integral inequalities for a variety of functions. For example, multiple authors in [37–45] gave the H-H inequalities and their right-left estimates for convex and co-ordinated convex functions via *π*1*Dq*, *π*2 *Dq*-derivatives and *qπ*1 , *qπ*2 -integrals. In the setting of *q*-calculus, Noor et al. [46] employed preinvexity to verify H-H inequalities. Nwaeze et al. [47] discovered several parameterized *q*-integral inequalities for generalized quasi-convex functions. In [48], Khan et al. used the concept of Green functions to develop some novel H-H type inequalities. In the context of *q*-calculus, Budak et al. [49], Ali et al. [50,51], and Vivas-Cortez et al. [52] demonstrated new boundaries for Simpson's and Newton's type inequalities for convex and coordinated convex functions. For *q*-Ostrowski inequality for convex and co-ordinated convex functions, see [53–55]. The authors used the *π*1*Dp*,*q*difference operator and the (*p*, *q*)*π*1 -integral to generalize the results of [39] and show H-H type inequalities and their left estimates [56]. The authors recently established the right estimates of H-H type inequalities shown by Kunt et al. [56] in [57]. Reference [36] can be used to solve (*p*, *q*)-Ostrowski type inequalities. The findings in [58] are a generalization of [33].

Inspired by the ongoing studies, we give the generalizations of the results proved in [33,39,41,59] by proving H-H trapezoid and midpoint type inequalities for preinvex functions using the concepts of (*p*, *q*)-difference operators and (*p*, *q*)-integral.

This paper is organized in the following way: Section 2 introduces the basics of *q*calculus and discusses other related research in the field. (*p*, *q*)-derivatives and integrals are discussed in Section 3. In Section 4, we show that in the (*p*, *q*)-calculus setting, H-H type inequalities exist for preinvex functions. Sections 5 and 6 prove new midpoint and trapezoid type inequalities for differentiable preinvex functions via (*p*, *q*)-calculus. The link between the findings reported here and analogous findings in the literature is also taken into account. Section 7 summarize the findings and suggests research topics for the future.

#### **2. Quantum Derivatives and Integrals**

This section discusses the key concepts and findings that will be needed to prove our critical findings in the next sections.

**Definition 1** ([7,9])**.** *A set ω* ⊆ R*n is known as invex with respect to the given η* : R*n* × R*n* → R*n if*

$$
\varkappa \varkappa + t\eta(\gamma, \varkappa) \in \omega, \forall \; \varkappa, \gamma \in \omega, t \in [0, 1].
$$

*The η-connected set is a more frequent name for the invex set ω.*

**Definition 2** ([7,9])**.** *Consider an invex set ω* ⊆ R*n with respect to η* : R*n* × R*n* → R*n*. *A mapping* Π : *ω* → R *is called preinvex, if*

$$\Pi(\varkappa + t\mathfrak{u}(\lambda^{\prime}\varkappa)) \le t\Pi(\lambda) + (1 - t)\Pi(\varkappa), \forall \varkappa, \gamma \in \omega, t \in [0, 1]. \tag{2}$$

*If* −Π *is preinvex, the mapping* Π *is called preconcave.*

**Remark 1.** *Definition 2 becomes the definition of convex functions if η*(*γ*,κ) = *γ* − κ *is set in Definition 2:*

Π(κ + *t*(*γ* − κ)) ≤ *t*Π(*γ*) + (1 − *t*)Π(κ)*,* ∀ κ, *γ* ∈ *ω, t* ∈ [0, 1].

**Condition C.** [9] The function *η* satisfies the following condition if

$$\begin{aligned} \eta(\gamma, \gamma + t\eta(\varkappa, \gamma)) &= -t\eta(\varkappa, \gamma), \\ \eta(\varkappa, \gamma + \eta(\varkappa, \gamma)) &= (1 - t)\eta(\varkappa, \gamma) \end{aligned} \tag{3}$$

for every κ, *γ* ∈ *ω* and any *t* ∈ [0, 1]. Note that for every κ, *γ* ∈ *ω*, *t*1, *t*2 ∈ [0, 1], and from Condition C, we have the following:

$$
\eta(\gamma + t\_2 \eta(\varkappa, \gamma), \gamma + t\_1 \eta(\varkappa, \gamma)) = (t\_2 - t\_1)\eta(\varkappa, \gamma).
$$

**Theorem 1** ([60] (Jensen's inequality for preinvex functions))**.** *Let* Π : *ω* → R *be a preinvex function. Let γ*1, *γ*2, ... , *γn* ∈ [0, 1] *be the coefficients such that n* ∑ *i*=0 *γi* = 1*, and let t*1, *t*2, ... , *tn* ∈ [0, 1] *be the coefficients. Then, the inequality*

$$\Pi\left(\sum\_{i=1}^{n} \gamma\_i(\varkappa + t\_i \eta(\gamma, \varkappa))\right) \le \sum\_{i=1}^{n} \gamma\_i \Pi(\varkappa + t\_i \eta(\gamma, \varkappa))\tag{4}$$

*holds for all* κ, *γ* ∈ *ω*.

Set the following notation [26]:

$$[n]\_q = \frac{1 - q^n}{1 - q} = 1 + q + q^2 + \dots + q^{n-1}, \ q \in (0, 1).$$

The *q*-Jackson integral of a mapping Π from 0 to *π*2 is given by Jackson [30] which is defined as:

$$\int\_0^{\pi\_2} \Pi(\varkappa) \, d\_{\overline{q}} \varkappa \quad = (1 - q)\pi\_2 \sum\_{n=0}^{\infty} q^n \Pi(\pi\_2 q^n), \text{ where } 0 < q < 1\tag{5}$$

assuming that the sum is absolutely convergent. Moreover, over the interval [*π*1, *π*2], he gave the following integral of a mapping Π:

$$\int\_{\frac{\pi\_1}{2}}^{\pi\_2} \Pi(\varkappa) \, d\_q \varkappa \, \, \, = \int\_0^{\pi\_2} \Pi(\varkappa) \, d\_q \varkappa \, \, \, - \int\_0^{\pi\_1} \Pi(\varkappa) \, \, d\_q \varkappa \, \, \, \,.$$

**Definition 3** ([32])**.** *The qπ*1 *-derivative of mapping* Π : [*π*1, *π*2] → R *is defined as:*

$${}\_{\pi\_1}D\_q\Pi(\varkappa)\;=\frac{\Pi(\varkappa)-\Pi(q\varkappa+(1-q)\pi\_1)}{(1-q)(\varkappa-\pi\_1)},\;\varkappa\neq\pi\_1.\tag{6}$$

*For* κ = *π*1*, we state π*1*Dq*Π(*π*1) = limκ→*π*1 *π*1*Dq*Π(κ) *if it exists and it is finite.*

**Definition 4** ([33])**.** *The qπ*2 *-derivative of mapping* Π : [*π*1, *π*2] → R *is given as:*

$${}^{\pi\_2}D\_q\Pi(\varkappa)\;=\frac{\Pi(q\varkappa+(1-q)\pi\_2)-\Pi(\varkappa)}{(1-q)(\pi\_2-\varkappa)},\;\varkappa\neq\pi\_2.\tag{7}$$

*For* κ = *π*2*, we state π*2*Dq*Π(*π*2) = limκ→*π*2 *π*2*Dq*Π(κ) *if it exists and it is finite.*

**Definition 5** ([32])**.** *The qπ*1 *-definite integral of mapping* Π : [*π*1, *π*2] → R *on* [*π*1, *π*2] *is defined as:*

$$\int\_{\pi\_1}^{\varkappa} \Pi(\tau) \, \, \_{\pi\_1} d\_{\eta} \tau \;= (1 - q)(\varkappa - \pi\_1) \sum\_{n=0}^{\infty} q^n \Pi(q^n \varkappa + (1 - q^n)\pi\_1), \\ \varkappa \in [\pi\_1, \pi\_2]. \tag{8}$$

Bermudo et al. [33], on the other hand, state the following concept of the *q*-definite integral:

**Definition 6** ([33])**.** *The qπ*2 *-definite integral of mapping* Π : [*π*1, *π*2] → R *on* [*π*1, *π*2] *is given as:*

$$\int\_{\varkappa}^{\pi\_2} \Pi(\tau) \,^{\pi\_2} d\_q \tau \;= (1 - q)(\pi\_2 - \varkappa) \sum\_{n=0}^{\infty} q^n \Pi(q^n \varkappa + (1 - q^n)\pi\_2), \; \varkappa \in [\pi\_1, \pi\_2]. \tag{9}$$

## **3. Post-Quantum Derivatives and Integrals**

We will go over some basic (*p*, *q*)-calculus concepts and notations in this section. The [*n*]*p*,*q* is said to be (*p*, *q*)-integers and expressed as:

$$\left[n\right]\_{p\mathcal{A}} = \frac{p^n - q^n}{p - q}$$

with 0 < *q* < *p* ≤ 1. The [*n*]*p*,*q*! and *n k* ! are called (*p*, *q*)-factorial and (*p*, *q*)-binomial, respectively, and expressed as:

$$\begin{aligned} \left[n\right]\_{p,q}! &= \prod\_{k=1}^n \left[k\right]\_{p,q'} \left[n \ge 1, \left[0\right]\_{p,q}! = 1, \ldots\right] \\ \left[\begin{array}{rcl} n & & & \left[n\right]\_{p,q}! \\ k & & & \left[\left[n-k\right]\_{p,q}! \left[k\right]\_{p,q}! \end{array} \right. \end{aligned}$$

**Definition 7** ([34])**.** *The* (*p*, *q*)*-derivative of mapping* Π : [*π*1, *π*2] → R *is given as:*

$$D\_{p,q} \Pi(\varkappa) = \frac{\Pi(p\varkappa) - \Pi(q\varkappa)}{(p-q)\varkappa}, \varkappa \neq 0 \tag{10}$$

*with* 0 < *q* < *p* ≤ 1.

**Definition 8** ([35])**.** *The* (*p*, *q*)*π*1 *-derivative of mapping* Π : [*π*1, *π*2] → R *is given as:*

$$D\_{\pi1}D\_{p,q}\Pi(\varkappa) = \frac{\Pi(p\varkappa + (1-p)\pi\_1) - \Pi(q\varkappa + (1-q)\pi\_1)}{(p-q)(\varkappa - \pi\_1)}, \varkappa \neq \pi\_1 \tag{11}$$

*with* 0 < *q* < *p* ≤ 1.

$$\text{For } \varkappa = \pi\_1 \text{, we state } \_{\pi\_1} D\_{p,q} \Pi(\pi\_1) = \lim\_{\varkappa \to \pi\_1} \_{\pi\_1} D\_{p,q} \Pi(\varkappa) \text{ if it exists and it is finite.}$$

**Definition 9** ([36])**.** *The* (*p*, *q*) *π*2 *-derivative of mapping* Π : [*π*1, *π*2] → R *is given as:*

$${}^{\pi\_2}D\_{p,q}\Pi(\varkappa) = \frac{\Pi(q\varkappa + (1-q)\pi\_2) - \Pi(p\varkappa + (1-p)\pi\_2)}{(p-q)(\pi\_2 - \varkappa)}, \varkappa \neq \pi\_2. \tag{12}$$

For κ = *π*2, we state *π*2*Dp*,*q*Π(*π*2) = limκ→*π*2 *π*2*Dp*,*q*Π(κ) if it exists and it is finite.

**Remark 2.** *It is clear that if we use p* = 1 *in (11) and (12), then the equalities (11) and (12) reduce to (6) and (7), respectively.*

**Definition 10** ([35])**.** *The definite (p*, *q)π*1 *-integral of mapping* Π : [*π*1, *π*2] → R *on* [*π*1, *π*2] *is stated as:*

$$\int\_{\pi\_1}^{\varkappa} \Pi(\tau) \, \_{\pi\_1} d\_{p,q} \tau = (p-q)(\varkappa - \pi\_1) \sum\_{n=0}^{\infty} \frac{q^n}{p^{n+1}} \Pi \left( \frac{q^n}{p^{n+1}} \varkappa + \left( 1 - \frac{q^n}{p^{n+1}} \right) \pi\_1 \right) \tag{13}$$

*with* 0 < *q* < *p* ≤ 1.

**Definition 11** ([36])**.** *The definite (p*, *q) π*2 *-integral of mapping* Π : [*π*1, *π*2] → R *on* [*π*1, *π*2] *is stated as:*

$$\int\_{\mathcal{X}}^{\pi\_2} \Pi(\pi) \,^{\pi\_2} d\_{p,q} \pi = (p-q)(\pi\_2 - \varkappa) \sum\_{n=0}^{\infty} \frac{q^n}{p^{n+1}} \Pi\left(\frac{q^n}{p^{n+1}} \varkappa + \left(1 - \frac{q^n}{p^{n+1}}\right) \pi\_2\right) \tag{14}$$

*with* 0 < *q* < *p* ≤ 1.

**Remark 3.** *It is evident that if we pick p* = 1 *in (13) and (14), then the equalities (13) and (14) change into (8) and (9), respectively.*

**Remark 4.** *If we take π*1 = 0 *and* κ = *π*2 = 1 *in (13), then we have*

$$\int\_0^1 \Pi(\tau) \, \_0d\_{p,q}\tau = (p-q) \sum\_{n=0}^\infty \frac{q^n}{p^{n+1}} \Pi\left(\frac{q^n}{p^{n+1}}\right).$$

*Similarly, by taking* κ = *π*1 = 0 *and π*2 = 1 *in (14), then we obtain that*

$$\int\_0^1 \Pi(\tau) \,^1 d\_{p,q} \tau = (p-q) \sum\_{n=0}^\infty \frac{q^n}{p^{n+1}} \Pi\left(1 - \frac{q^n}{p^{n+1}}\right).$$

In [56], Kunt et al. proved the following H-H type inequalities for convex functions via the (*p*, *q*)*π*1 -integral:

**Theorem 2.** *For a convex mapping* Π : [*π*1, *π*2] → R, *which is differentiable on* [*π*1, *π*2], *the following inequalities hold for the* (*p*, *q*)*π*1 *-integral:*

$$\Pi\left(\frac{q\pi\_1 + p\pi\_2}{[2]\_{p,q}}\right) \le \frac{1}{p(\pi\_2 - \pi\_1)} \int\_{\pi\_1}^{p\pi\_2 + (1-p)\pi\_1} \Pi(\varkappa) \, \pi\_1 \, d\_{p,q}\varkappa \le \frac{q\Pi(\pi\_1) + p\Pi(\pi\_2)}{[2]\_{p,q}}\tag{15}$$

$$\text{where } 0 < q < p \le 1.$$

**Lemma 1** ([58])**.** *We have the following equalities:*

$$\int\_{\pi\_1}^{\pi\_2} (\pi\_2 - \varkappa)^{\alpha \cdot \pi\_2} d\_{p,q} \varkappa = \frac{(\pi\_2 - \pi\_1)^{\alpha + 1}}{[\alpha + 1]\_{p,q}}.$$

$$\int\_{\pi\_1}^{\pi\_2} (\varkappa - \pi\_1)^{\kappa} \,\_{\pi\_1} d\_{p,q} \varkappa = \frac{(\pi\_2 - \pi\_1)^{\kappa + 1}}{[\alpha + 1]\_{p,q}}$$

*where α* ∈ R − {−1}.

#### **4. New H-H Type Inequalities for Post-Quantum Integrals**

We present a new variant of the (*p*, *q*)-H-H inequality for preinvex functions in this section. It is also demonstrated that the results presented here are a generalization of some previously published results. For brevity, we use *I* = [*π*2 + *η*(*π*1, *π*2), *π*2] and *J* = [*π*1, *π*1 + *η*(*π*2, *π*1)].

**Theorem 3.** *For a differentiable preinvex mapping* Π : *I* → R, *the following inequality holds for the* (*p*, *q*) *π*2 *-integral:*

$$\begin{split} \Pi \left( \frac{p\eta\left(\pi\_{1},\pi\_{2}\right) + \left[2\right]\_{p,q}\pi\_{2}}{\left[2\right]\_{p,q}} \right) & \leq \ \frac{1}{p\eta\left(\pi\_{2},\pi\_{1}\right)} \int\_{\pi\_{2} + p\eta\left(\pi\_{1},\pi\_{2}\right)}^{\pi\_{2}} \Pi\left(\varkappa\right)^{\pi\_{2}} d\_{p,q} \varkappa \\ & \leq \ \ \frac{p\Pi\left(\pi\_{2} + p\eta\left(\pi\_{1},\pi\_{2}\right)\right) + q\Pi\left(\pi\_{2}\right)}{\left[2\right]\_{p,q}}, \end{split} \tag{16}$$

*where* 0 < *q* < *p* ≤ 1.

**Proof.** For preinvex functions, we can use Jensen's inequality

$$\Pi \left( \frac{1}{p\eta(\pi\_2, \pi\_1)} \int\_{\pi\_2 + p\eta(\pi\_1, \pi\_2)}^{\pi\_2} \varkappa^{\pi\_2} d\_{p, \emptyset} \varkappa \right) \le \frac{1}{p\eta(\pi\_2, \pi\_1)} \int\_{\pi\_2 + p\eta(\pi\_1, \pi\_2)}^{\pi\_2} \Pi(\varkappa)^{\pi\_2} d\_{p, \emptyset} \varkappa$$

and from the Definition 11, one can easily observe that

$$\frac{1}{p\eta(\,\pi\_2,\,\pi\_1)}\int\_{\pi\_2+p\eta(\,\pi\_1,\,\pi\_2)}^{\pi\_2} \varkappa^{\pi\_2}d\_{p,q}\varkappa = \frac{p\eta(\,\pi\_1,\,\pi\_2)+[2]\_{p,q}\pi\_2}{[2]\_{p,q}}$$

Thus, the first part of the inequality (16) is proved. To prove the second inequality in (16), first, we note that Π is preinvex function and we have

$$
\Pi(\pi\_2 + t\eta(\pi\_1, \pi\_2)) \le t\Pi(\pi\_1) + (1 - t)\Pi(\pi\_2). \tag{17}
$$

.

Applying the (*p*, *q*) *π*2 -integral on the both sides of (17), we have

$$\frac{1}{p\eta\left(\pi\_{2},\pi\_{1}\right)}\int\_{\pi\_{2}+p\eta\left(\pi\_{1},\pi\_{2}\right)}^{\pi\_{2}}\Pi(\varkappa)^{\cdot\pi\_{2}}d\_{p,q}\varkappa \leq \frac{p\Pi(\pi\_{2}+p\eta\left(\pi\_{1},\pi\_{2}\right))+q\Pi(\pi\_{2})}{[2]\_{p,q}}.$$

Hence, the proof is completed.

**Remark 5.** *We obtain Theorem 5 in [59], by letting p* = 1 *in Theorem 3.*

**Remark 6.** *If we set p* = 1 *in Theorem 3 and later assume that η*(*π*2, *π*1) = −*η*(*π*1, *π*2) = *π*2 − *π*1*, then Theorem 3 becomes Theorem 12 in [33].*

**Example 1.** *Let* Π(κ) = −|κ|*. Then,* Π *is preinvex function with respect to the following bifunction:*

$$\eta(\varkappa, y) = \begin{cases} \varkappa - y, \text{ if } \varkappa y \ge 0 \\ y - \varkappa, \text{ if } \varkappa y \le 0. \end{cases}$$

*1. Let us consider π*1, *π*2 > 0, *then η*(*π*1, *π*2) = *π*1 − *π*2 *and*

$$\begin{aligned} \Pi \left( \frac{p\eta(\pi\_1, \pi\_2) + [2]\_{p,q}\pi\_2}{[2]\_{p,q}} \right) &=& -\frac{p\pi\_1 + q\pi\_2}{[2]\_{p,q}},\\ \frac{1}{p\eta(\pi\_2, \pi\_1)} \int\_{\pi\_2 + p\eta(\pi\_1, \pi\_2)}^{\pi\_2} \Pi(\varkappa)^{\pi\_2} d\_{p,q}\varkappa &=& -\frac{p\pi\_1 + q\pi\_2}{[2]\_{p,q}} \end{aligned}$$

*and*

$$\frac{p\Pi(\pi\_2 + p\eta(\pi\_1, \pi\_2)) + q\Pi(\pi\_2)}{[2]\_{p,q}} = -\frac{p\pi\_1 + q\pi\_2}{[2]\_{p,q}}$$

.

.

$$\text{12. }\quad \text{Let } \pi\_1, \pi\_2 < 0. \text{ Then, } \eta(\pi\_1, \pi\_2) = \pi\_1 - \pi\_2 \text{ and}$$

$$\begin{aligned} \Pi \Big( \frac{p\eta\left(\pi\_1,\pi\_2\right) + \left[2\right]\_{p,q}\pi\_2}{\left[2\right]\_{p,q}} \Big) &=& \frac{p\pi\_1 + q\pi\_2}{\left[2\right]\_{p,q}},\\ \frac{1}{p\eta\left(\pi\_2,\pi\_1\right)} \int\_{\pi\_2 + p\eta\left(\pi\_1,\pi\_2\right)}^{\pi\_2} \Pi\left(\varkappa\right)^{\pi\_2} d\_{p,q} \varkappa &=& \frac{p\pi\_1 + q\pi\_2}{\left[2\right]\_{p,q}} \end{aligned}$$

*and*

$$\frac{p\Pi(\pi\_2 + p\eta(\pi\_1, \pi\_2)) + q\Pi(\pi\_2)}{[2]\_{p,q}} = -\frac{p\pi\_1 + q\pi\_2}{[2]\_{p,q}}$$

*3. Finally, let π*1 < 0 < *π*2*. Then, η*(*π*1, *π*2) = *π*2 − *π*1 *and*

$$\begin{aligned} \Pi \left( \frac{p\eta(\pi\_1, \pi\_2) + [2]\_{p,q}\pi\_2}{[2]\_{p,q}} \right) &=& -\frac{(2p+q)\pi\_2 - p\pi\_1}{[2]\_{p,q}},\\ \frac{1}{p\eta(\pi\_2, \pi\_1)} \int\_{\pi\_2 + p\eta(\pi\_1, \pi\_2)}^{\pi\_2} \Pi(\varkappa) \, \prescript{\pi\_2}{}{}\_{p,q}\varkappa &=& -\frac{(2p+q)\pi\_2 - p\pi\_1}{[2]\_{p,q}} \end{aligned}$$

*and*

$$\frac{p\Pi(\pi\_2 + p\eta(\pi\_1, \pi\_2)) + q\Pi(\pi\_2)}{[2]\_{p,q}} = -\frac{(2p+q)\pi\_2 - p\pi\_1}{[2]\_{p,q}}.$$

*It is clear that the Theorem 3 is valid.*

**Theorem 4.** *For a differentiable preinvex mapping* Π : *J* → R, *the following inequality holds for* (*p*, *q*)*π*1 *-integral:*

$$\begin{split} \Pi \left( \frac{p\eta\left(\left.\pi\_{2},\pi\_{1}\right)+\left[2\right]\_{p,q}\pi\_{1}}{\left[2\right]\_{p,q}} \right) & \leq \ \frac{1}{p\eta\left(\left.\pi\_{2},\pi\_{1}\right)} \int\_{\pi\_{1}}^{\pi\_{1}+p\eta\left(\left.\pi\_{2},\pi\_{1}\right)} \Pi\left(\varkappa\right)\right.\right.} \Pi\left(\varkappa\right) \left.\pi\_{1}\right|d\_{p,q}\varkappa \\ & \leq \ \ \frac{p\Pi\left(\left.\pi\_{1}\right)+q\Pi\left(\left.\pi\_{1}+\eta\left(\left.\pi\_{2},\pi\_{1}\right)\right)\right.}{\left[2\right]\_{p,q}} \right. \end{split} \tag{18}$$

*where* 0 < *q* < *p* ≤ 1.

**Proof.** One can easily obtain the inequality (18) by following the methodology used in the proof of Theorem 3 and taking into account Definition 10 of the (*p*, *q*)*π*1 -integral.

**Remark 7.** *We obtain Theorem 6 in [59] by letting p* = 1 *in Theorem 4.*

**Remark 8.** *If we set p* = 1 *in Theorem 4 and later assume that η*(*π*2, *π*1) = *π*2 − *π*1*, then Theorem 4 reduces to Theorem 6 in [39].*

#### **5. Midpoint Type Inequalities through (***p***,** *q***)** *π***2 -Integral**

We present some new midpoint type inequalities using the (*p*, *q*)-derivative and integral in this section.

The following crucial lemma is required to prove the main results of this section.

**Lemma 2.** *Let* Π : *I* → R *be a differentiable function on I*◦*. If π*2*Dp*,*q*Π *is continuous and integrable on I, then we have the following identity:*

$$\eta\left(\pi\_{2},\pi\_{1}\right)\left[\int\_{0}^{\frac{p}{|\pi\_{p,q}|}}qt^{\pi\_{2}}D\_{p,q}\Pi\left(t\left(\pi\_{2}+\eta\left(\pi\_{1},\pi\_{2}\right)\right)+(1-t)\pi\_{2}\right)d\_{p,q}t\right]$$

$$+\int\_{\frac{p}{|\pi\_{p,q}|}}^{1}(qt-1)^{\pi\_{2}}D\_{p,q}\Pi\left(t\left(\pi\_{2}+\eta\left(\pi\_{1},\pi\_{2}\right)\right)+(1-t)\pi\_{2}\right)d\_{p,q}t\right]$$

$$=\frac{1}{p\eta\left(\pi\_{2},\pi\_{1}\right)}\int\_{\pi\_{2}+p\eta\left(\pi\_{1},\pi\_{2}\right)}^{\pi\_{2}}\Pi\left(\varkappa\right)^{\pi\_{2}}d\_{p,q}\varkappa-\Pi\left(\frac{p\eta\left(\pi\_{1},\pi\_{2}\right)+[2]\_{p,q}\pi\_{2}}{[2]\_{p,q}}\right),\tag{19}$$

*where* 0 < *q* < *p* ≤ 1.

**Proof.** From Definition 9, we have

$$\begin{aligned} &\frac{\pi\_2 D\_{p,q} \Pi(t(\pi\_2 + \eta(\pi\_1, \pi\_2)) + (1 - t)\pi\_2)}{\Pi(qt(\pi\_2 + \eta(\pi\_1, \pi\_2)) + (1 - qt)\pi\_2) - \Pi(pt(\pi\_2 + \eta(\pi\_1, \pi\_2)) + (1 - pt)\pi\_2)}. \end{aligned} \tag{20}$$

From the left side of equality (19), we have

$$\begin{split} &\eta(\pi\_{2},\pi\_{1})\left[\int\_{0}^{\frac{p}{|\pi\_{p,q}|}} q t^{\frac{\pi\_{2}}{\pi\_{2}}} D\_{p,q} \Pi(t(\pi\_{2} + \eta(\pi\_{1},\pi\_{2})) + (1-t)\pi\_{2}) d\_{p,q} t \\ &\quad + \int\_{\frac{p}{|\pi\_{p,q}|}}^{1} (qt - 1)^{\frac{\pi\_{2}}{\pi\_{2}}} D\_{p,q} \Pi(t(\pi\_{2} + \eta(\pi\_{1},\pi\_{2})) + (1-t)\pi\_{2}) d\_{p,q} t \right] \\ &= \quad \eta(\pi\_{2},\pi\_{1})\left[\int\_{0}^{\frac{p}{|\pi\_{p,q}|}} \pi\_{2} D\_{p,q} \Pi(t(\pi\_{2} + \eta(\pi\_{1},\pi\_{2})) + (1-t)\pi\_{2}) d\_{p,q} t \\ &\quad + \int\_{0}^{1} q t^{\frac{\pi\_{2}}{\pi\_{2}}} D\_{p,q} \Pi(t(\pi\_{2} + \eta(\pi\_{1},\pi\_{2})) + (1-t)\pi\_{2}) d\_{p,q} t \\ &\quad - \int\_{0}^{1} \pi\_{2} D\_{p,q} \Pi(t(\pi\_{2} + \eta(\pi\_{1},\pi\_{2})) + (1-t)\pi\_{2}) d\_{p,q} t \right]. \tag{21} \end{split}$$

By the equality (14), we have

$$\begin{split} &\quad \int\_{0}^{\frac{p}{\left[\Xi\right]\_{p,q}}} \, \, \_2D\_{p,q} \Pi(t(\pi\_2 + \eta(\pi\_1, \pi\_2)) + (1-t)\pi\_2) d\_{p,q} t \\ &= \quad \frac{1}{\eta(\pi\_2, \pi\_1)(p-q)} \int\_{0}^{\frac{p}{\left[\Xi\right]\_{p,q}}} \frac{\Pi(qt(\pi\_2 + \eta(\pi\_1, \pi\_2)) + (1-qt)\pi\_2)}{-\Pi(pt(\pi\_2 + \eta(\pi\_1, \pi\_2)) + (1-pt)\pi\_2)} d\_{p,q} t \\ &= \quad \frac{1}{\eta(\pi\_2, \pi\_1)} \left[ \sum\_{n=0}^{\infty} \Pi\left(\frac{p}{\left[2\right]\_{p,q}} \frac{q^{n+1}}{p^{n+1}} (\pi\_2 + \eta(\pi\_1, \pi\_2)) + \left(1 - \frac{p}{\left[2\right]\_{p,q}} \frac{q^{n+1}}{p^{n+1}}\right) \pi\_2\right) \right. \\ &\left. - \sum\_{n=0}^{\infty} \Pi\left(\frac{p}{\left[2\right]\_{p,q}} \frac{q^n}{p^n} (\pi\_2 + \eta(\pi\_1, \pi\_2)) + \left(1 - \frac{p}{\left[2\right]\_{p,q}} \frac{q^n}{p^n}\right) \pi\_2\right) \right] \end{split}$$

$$=\frac{\Pi(\pi\_2)}{\eta(\pi\_2,\pi\_1)} - \frac{1}{\eta(\pi\_2,\pi\_1)}\Pi\left(\frac{p\eta(\pi\_1,\pi\_2) + [2]\_{p,q}\pi\_2}{[2]\_{p,q}}\right),\tag{22}$$

$$\begin{split} &\int\_0^1 \pi^{\pi\_2} D\_{p,q} \Pi(t(\pi\_2 + \eta(\pi\_1,\pi\_2)) + (1-t)\pi\_2) d\_{p,q}t \\ &=\frac{1}{\eta(\pi\_2,\pi\_1)(p-q)}\int\_0^1 \frac{\Pi(qt(\pi\_2 + \eta(\pi\_1,\pi\_2)) + (1-qt)\pi\_2)}{t} d\_{p,q}t \\ &=\frac{1}{\eta(\pi\_2,\pi\_1)}\left[\sum\_{n=0}^\infty \Pi\left(\frac{q^{n+1}}{p^{n+1}}(\pi\_2 + \eta(\pi\_1,\pi\_2)) + \left(1 - \frac{q^{n+1}}{p^{n+1}}\right)\pi\_2\right)\right] \\ &=\frac{\Pi(\pi\_2) - \Pi(\pi\_2 + \eta(\pi\_1,\pi\_2))}{\eta(\pi\_2,\pi\_1)} \end{split} \tag{23}$$

and

 1 0 *t π*2*Dp*,*q*Π(*t*(*π*2 + *η*(*π*1, *π*2)) + (1 − *t*)*π*2)*dp*,*qt* = 1 *η*(*π*2, *π*1)(*p* − *q*) 1 0 Π(*qt*(*π*2 + *η*(*π*1, *π*2)) + (1 − *qt*)*π*2) −Π(*pt*(*π*2 + *η*(*π*1, *π*2)) + (1 − *pt*)*π*2) *dp*,*qt* = 1 *η*(*π*2, *π*1) ⎡ ⎢ ⎢ ⎣ ∞ ∑ *n*=0 *qn pn*+1 Π *qn*+1 *pn*+1 (*π*2 + *η*(*π*1, *π*2)) + 1 − *qn*+1 *pn*+1 *π*2 − ∞ ∑ *n*=0 *qn pn*+1 Π *qn pn* (*π*2 + *η*(*π*1, *π*2)) + 1 − *qn pn π*2 ⎤ ⎥ ⎥ ⎦ = 1 *η*(*π*2, *π*1) ⎡ ⎢ ⎢ ⎣ 1 *q* ∞ ∑ *n*=0 *qn*+1 *pn*+1 Π *qn*+1 *pn*+1 (*π*2 + *η*(*π*1, *π*2)) + 1 − *qn*+1 *pn*+1 *π*2 − 1 *p* ∞ ∑ *n*=0 *qn pn* Π *qn pn* (*π*2 + *η*(*π*1, *π*2)) + 1 − *qn pn π*2 ⎤ ⎥ ⎥ ⎦ = 1 *η*(*π*2, *π*1) ⎡ ⎣ 1 *q* − 1 *p* ∞ ∑ *n*=0 *qn pn* Π *qn pn* (*π*2 + *η*(*π*1, *π*2)) + 1 − *qn pn π*2 −1 *q* Π((*π*2 + *η*(*π*1, *π*2))) ⎤ ⎦ = 1 *η*(*π*2, *π*1) *p* − *q pq* ∞ ∑ *n*=0 *qn pn* Π - *qn pn* (*π*2 + *η*(*π*1, *π*2)) + - 1 − *qn pn π*2 −1 *q* Π(*π*2 + *η*(*π*1, *π*2)) = 1 *η*(*π*2, *π*1) 1 *pqη*(*π*2, *π*1) *π*2 *π*2+*pη*(*π*1,*π*2) Π(κ) *π*2 *dp*,*q*κ − 1 *q* Π(*π*2 + *η*(*π*1, *π*2)) . (24)

By using (22)–(24) in (21), we obtain the desired identity (19). Thus, the proof is completed.

**Remark 9.** *We obtain Lemma 3 in [59] by letting p* = 1 *in Lemma 2.*

**Remark 10.** *If we use p* = 1 *in Lemma 2 and later consider η*(*π*2, *π*1) = −*η*(*π*1, *π*2) = *π*2 − *π*1*, then Lemma 2 becomes Lemma 2 in [41].*

**Theorem 5.** *Suppose that the assumptions of Lemma 2 hold. If π*2*Dp*,*q*Π *is a preinvex function over I, then we have the following new inequality:*

$$\begin{array}{c} \left| \frac{1}{p\eta(\pi\_2,\pi\_1)} \int\_{\pi\_2+p\eta(\pi\_1,\pi\_2)}^{\pi\_2} \Pi(\varkappa)\,^{\pi\_2}d\_{p,q}\varkappa - \Pi\left(\frac{p\eta(\pi\_1,\pi\_2)+[2]\_{p,q}\pi\_2}{[2]\_{p,q}}\right) \right| \\ \leq \quad \eta\left(\pi\_2,\pi\_1\right) \left[ \left( \left| \,^{\pi\_2}D\_{p,\emptyset}\Pi(\pi\_1) \right| A\_1(p,q) + \left| \,^{\pi\_2}D\_{p,\emptyset}\Pi(\pi\_2) \right| A\_2(p,q) \right) \\ \quad + \left( \left| \,^{\pi\_2}D\_{p,\emptyset}\Pi(\pi\_1) \right| A\_3(p,q) + \left| \,^{\pi\_2}D\_{p,\emptyset}\Pi(\pi\_2) \right| A\_4(p,q) \right) \right] \end{array} \tag{25}$$

*where*

$$\begin{array}{rcl} A\_1(p,q) &=& \frac{qp^3}{[2]\_{p,q}^3[3]\_{p,q}},\\ A\_2(p,q) &=& \frac{q\left(p^3(p^2+q^2-p)+p^2[3]\_{p,q}\right)}{[2]\_{p,q}^4[3]\_{p,q}},\\ A\_3(p,q) &=& \frac{q(q+2p)}{[2]\_{p,q}} - \frac{q^2(q^2+3p^2+3pq)}{[2]\_{p,q}^3[3]\_{p,q}},\\ A\_4(p,q) &=& \frac{q}{[2]\_{p,q}} - \frac{q^2(q+2p)}{[2]\_{p,q}^4} - A\_3(p,q). \end{array}$$

**Proof.** Taking the modulus in Lemma 2 and using the preinvexity of *π*2*Dp*,*q*Π , we obtain that

$$\begin{split} & \left| \frac{1}{p\eta(\pi\_{2},\pi\_{1})} \int\_{\pi\_{2}+p\eta(\pi\_{1},\pi\_{2})}^{\pi\_{2}} \Pi(\varkappa) \, ^{\pi\_{2}} d\_{p,q} \varkappa - \Pi\left(\frac{p\eta(\pi\_{1},\pi\_{2})+[2]\_{p,q}\pi\_{2}}{[2]\_{p,q}}\right) \right| \\ & \leq \quad \eta(\pi\_{2},\pi\_{1}) \left[ \int\_{0}^{\frac{p}{\left|\pi\_{1}\right|q}} q\left( ^{\pi\_{2}}D\_{p,q}\Pi(t(\pi\_{2}+\eta(\pi\_{1},\pi\_{2}))+(1-t)\pi\_{2}) \big| d\_{p,q}t \right. \\ & \left. + \int\_{\frac{p}{\left|\pi\_{2}\right|q}}^{\frac{p}{\left|\pi\_{2}\right|q}} (1-qt) \left| \, ^{\pi\_{2}}D\_{p,q}\Pi(t(\pi\_{2}+\eta(\pi\_{1},\pi\_{2}))+(1-t)\pi\_{2}) \big| d\_{p,q}t \right| \\ & \leq \quad \eta(\pi\_{2},\pi\_{1}) \left[ q\left( \int\_{0}^{\frac{p}{\left|\pi\_{1}\right|q}} t\left( {t}^{\left|\pi\_{2}\right|}D\_{p,q}\Pi(\pi\_{1}) \right|+(1-t) \left| {t}^{\left|\pi\_{2}\right|}D\_{p,q}\Pi(\pi\_{2}) \right| \right) d\_{p,q}t \right. \\ & \left. + \int\_{\frac{p}{\left|\pi\_{2}\right|q}}^{\frac{p}{\left|\pi\_{2}\right|q}} (1-qt) \left( {t}^{\left|\$$

One can easily compute the integrals that appeared on the right side of the inequality (26)

$$\int\_0^{\frac{p}{\left[2\right]\_{p,q}}} t^2 d\_{p,q} t \quad = \quad \frac{p^3}{\left[2\right]\_{p,q}^3 \left[3\right]\_{p,q}} \, ^\prime \tag{27}$$

$$\int\_{0}^{\frac{p}{\left[\frac{p}{2}\right]\_{p,q}}} t(1-t) d\_{p,q}t \quad = \frac{p^3(p^2+q^2-p) + p^2[3]\_{p,q}}{[2]\_{p,q}^4[3]\_{p,q}},\tag{28}$$

$$\int\_{\frac{p}{\left[2\right]\_{p,q}}}^{1} t(1-qt)d\_{p,q}t \quad = \quad \frac{q(q+2p)}{\left[2\right]\_{p,q}} - \frac{q^2(q^2+3p^2+3pq)}{\left[2\right]\_{p,q}^{3}\left[3\right]\_{p,q}},\tag{29}$$

$$\int\_{\frac{p}{\left[2\right]\_{p,q}}}^{1} (1-t)(1-qt)d\_{p,q}t \quad = \frac{q}{\left[2\right]\_{p,q}} - \frac{q^2(q+2p)}{\left[2\right]\_{p,q}^3}$$

$$-\left(\frac{q(q+2p)}{[2]\_{p,q}} - \frac{q^2\left(q^2+3p^2+3pq\right)}{[2]\_{p,q}^3[3]\_{p,q}}\right).\tag{30}$$

Making use of (27)–(30) in (26), gives us the required inequality (25). Hence, the proof is finished.

**Remark 11.** *We obtain Theorem 9 in [59] by letting p* = 1 *in Theorem 5.*

**Remark 12.** *If we take p* = 1 *in Theorem 5 and later consider η*(*π*2, *π*1) = −*η*(*π*1, *π*2) = *π*2 − *π*1*, then Theorem 5 reduces to Theorem 5 in [41].*

**Theorem 6.** *Suppose that the assumptions of Lemma 2 hold. If π*2*Dp*,*q*Π *r , r* ≥ 1 *is a preinvex function over I, then we have the following new inequality:*

$$\begin{split} & \left| \frac{1}{p\eta(\pi\_{2},\pi\_{1})} \int\_{\pi\_{2}+p\eta(\pi\_{1},\pi\_{2})}^{\pi\_{2}} \Pi(\varkappa)^{\top 2} d\_{p,q} \varkappa - \Pi \left( \frac{p\eta(\pi\_{1},\pi\_{2}) + [2]\_{p,q} \pi\_{2}}{[2]\_{p,q}} \right) \right| \\ & \leq \quad \eta(\pi\_{2},\pi\_{1}) \left( \frac{p^{2}}{\left( [2]\_{p,q} \right)^{3}} \right)^{1 - \frac{1}{p}} \left[ \left\{ \left| \begin{smallmatrix} \pi\_{2} \boldsymbol{D}\_{p,q} \Pi(\pi\_{1}) \end{smallmatrix} \right| ^{\prime} \boldsymbol{A}\_{1}(p,q) + \left| \begin{smallmatrix} \pi\_{2} \boldsymbol{D}\_{p,q} \Pi(\pi\_{2}) \end{smallmatrix} \right| \operatorname{\boldsymbol{\varPi}}(\pi\_{2}) \right|^{\prime} \boldsymbol{A}\_{2}(p,q) \right\}^{\frac{1}{\prime}} \\ & \quad + \left\{ \left| \begin{smallmatrix} \pi\_{2} \boldsymbol{D}\_{p,q} \Pi(\pi\_{1}) \end{smallmatrix} \right| \operatorname{\boldsymbol{\upvarPi}}(\pi\_{1}) + \left| \begin{smallmatrix} \pi\_{2} \boldsymbol{D}\_{p,q} \Pi(\pi\_{2}) \end{smallmatrix} \right| \operatorname{\boldsymbol{\upvarPi}}(\pi\_{2}) \right|^{\prime} \boldsymbol{A}\_{2}(p,q) \right\}^{\frac{1}{\prime}}, \tag{31} \end{split} \tag{32}$$

*where A*1(*p*, *q*) − *A*4(*p*, *q*) *are given in Theorem 5.*

**Proof.** Taking the modulus in Lemma 2, applying the well-known power mean inequality for (*p*, *q*)-integrals and the preinvexity of *π*2*Dp*,*q*Π *r* , *r* ≥ 1, we have

 1 *pη*(*π*2, *π*1) *π*2 *π*2+*pη*(*π*1,*π*2) Π(κ) *π*2 *dp*,*q*κ − Π *pη*(*π*1, *π*2) + [2]*p*,*qπ*2 [2]*p*,*q* ≤ *η*(*π*2, *π*1) *p* [2]*p*,*q* 0 *qt π*2*Dp*,*q*Π(*t*(*π*2 + *η*(*π*1, *π*2)) + (1 − *t*)*π*2) *dp*,*qt* + 1 *p* [2]*p*,*q* (1 − *qt*) *π*2*Dp*,*q*Π(*t*(*π*2 + *η*(*π*1, *π*2)) + (1 − *t*)*π*2) *dp*,*qt* ⎤ ⎦ ≤ *η*(*π*2, *π*1) ⎡ ⎣ *p* [2]*p*,*q* 0 *qtdp*,*qt* 1− 1 *r* × *q p* [2]*p*,*q* 0 *t t π*2*Dp*,*q*Π(*π*1) *r* + (1 − *t*) *π*2*Dp*,*q*Π(*π*2) *r dp*,*qt* !1 *r* + ⎛ ⎝ 1 *p* [2]*p*,*q* (1 − *qt*)*dp*,*qt* ⎞ ⎠ 1− 1 *r* × ⎧ ⎨ ⎩ *q* 1 *p* [2]*p*,*q* (1 − *qt*) *t π*2*Dp*,*q*Π(*π*1) *r* + (1 − *t*) *π*2*Dp*,*q*Π(*π*2) *r dp*,*qt* ⎫ ⎬ ⎭ 1 *r* ⎤ ⎥ ⎦ = *η*(*π*2, *π*1) ⎛ ⎜⎝ *p*2 [2]*p*,*q* 3 ⎞ ⎟⎠ 1− 1 *r* ) *π*2*Dp*,*q*Π(*π*1) *r A*1(*p*, *q*) + *π*2*Dp*,*q*Π(*π*2) *r A*2(*p*, *q*) \*1 *r*

$$+ \left\{ \left| \,^{\pi\_2} D\_{p,q} \Pi(\pi\_1) \right| \,^{\mathsf{r}} A\_3(p,q) + \left| \,^{\pi\_2} D\_{p,q} \Pi(\pi\_2) \right| \,^{\mathsf{r}} A\_4(p,q) \right\}^{\frac{1}{r}} ]$$

which completes the proof.

**Remark 13.** *We obtain Theorem 10 in [59] by letting p* = 1 *in Theorem 6.*

**Remark 14.** *If we put p* = 1 *in Theorem 6 and later assume that η*(*π*2, *π*1) = −*η*(*π*1, *π*2) = *π*2 − *π*1*, then Theorem 6 reduces to Theorem 6 in [41].*

**Theorem 7.** *Suppose that the assumptions of Lemma 2 hold. If π*2*Dp*,*q*Π *r , r* > 1 *is a preinvex function over I, then we have the following new inequality:*

 1 *pη*(*π*2, *π*1) *π*2 *π*2+*pη*(*π*1,*π*2) Π(κ) *π*2 *dp*,*q*κ − Π *pη*(*π*1, *π*2) + [2]*p*,*qπ*2 [2]*p*,*q* ≤ *qη*(*π*2, *π*1) ⎡ ⎢ ⎢ ⎢ ⎣ ⎛ ⎝ *p* [2]*p*,*q s*+1- *p* − *q ps*+1 − *qs*+1 ⎞ ⎠ 1 *s* ⎧ ⎪⎪⎨ ⎪⎪⎩ *π*2*Dp*,*q*Π(*π*1) *r* - *p*2 [2] 3 *p*,*q* + *π*2*Dp*,*q*Π(*π*2) *r* - *p*3+*pq*2+2*p*2*q*−*p*2 [2] 3 *p*,*q* ⎫ ⎪⎪⎬ ⎪⎪⎭ 1 *r* + ⎛ ⎝ 1 *p* [2]*p*,*q* -1 *q* − *t s dp*,*qt* ⎞ ⎠ 1 *s* ⎧ ⎪⎪⎨ ⎪⎪⎩ *π*2*Dp*,*q*Π(*π*1) *r* -[2]*p*,*q*−*p*2 [2] 3 *p*,*q* + *π*2*Dp*,*q*Π(*π*2) *r q*[2] 2 *p*,*q*+*p*2−*p*−*q* [2] 3 *p*,*q* ⎫ ⎪⎪⎬ ⎪⎪⎭ 1 *r* ⎤ ⎥ ⎥ ⎥ ⎦ , (32)

*where s* + *r* = *sr*.

**Proof.** Taking the modulus in Lemma 2, by applying the well-known Hölder inequality for definite (*p*, *q*)-integrals and the preinvexity of *π*2*Dp*,*q*Π *r* , *r* > 1, we obtain that

 1 *pη*(*π*2, *π*1) *π*2 *π*2+*pη*(*π*1,*π*2) Π(κ) *π*2 *dp*,*q*κ − Π *pη*(*π*1, *π*2) + [2]*p*,*qπ*2 [2]*p*,*q* ≤ *qη*(*π*2, *π*1) *p* [2]*p*,*q* 0 *t π*2*Dp*,*q*Π(*t*(*π*2 + *η*(*π*1, *π*2)) + (1 − *t*)*π*2) *dp*,*qt* + 1 *p* [2]*p*,*q* -1 *q* − *t π*2*Dp*,*q*Π(*t*(*π*2 + *η*(*π*1, *π*2)) + (1 − *t*)*π*2) *dp*,*qt* ⎤ ⎦ ≤ *qη*(*π*2, *π*1) ⎡ ⎣ *p* [2]*p*,*q* 0 *t s dp*,*qt* 1 *s p* [2]*p*,*q* 0 *t π*2*Dp*,*q*Π(*π*1) *r* + (1 − *t*) *π*2*Dp*,*q*Π(*π*2) *r dp*,*qt* !1 *r* + ⎛ ⎝ 1 *p* [2]*p*,*q* -1 *q* − *t s dp*,*qt* ⎞ ⎠ 1 *s* × ⎧ ⎨ ⎩ 1 *p* [2]*p*,*q t π*2*Dp*,*q*Π(*π*1) *r* + (1 − *t*) *π*2*Dp*,*q*Π(*π*2) *r dp*,*qt* ⎫ ⎬ ⎭ 1 *r* ⎤ ⎥ ⎦. (33)

One can easily evaluate the integrals that appear on the right side of the inequality (33)

$$\left(\int\_0^{\frac{p}{\left[\frac{p}{2}\right]\_{p,q}}} t^s d\_{p,q} t\right)^{\frac{1}{s}} = \left(\left(\frac{p}{\left[\frac{p}{2}\right]\_{p,q}}\right)^{s+1} \left(\frac{p-q}{p^{s+1}-q^{s+1}}\right)\right)^{\frac{1}{s}}\tag{34}$$

$$\int \frac{p}{\left[\frac{p}{2}\right]\_{p,q}} \dots \qquad p^2\tag{35}$$

$$\int\_{0}^{\frac{\nu}{[2]\_{p,q}}} t d\_{p,q} t \quad = \quad \frac{p^2}{[2]\_{p,q}^3},\tag{35}$$

$$\int\_{0}^{\frac{p}{\left[\frac{p}{2}\right]\_{p,q}}} (1-t) d\_{p,q} t \quad = \frac{p^3 + pq^2 + 2p^2q - p^2}{\left[2\right]\_{p,q}^3},\tag{36}$$

$$\int\_{\frac{p}{\left[2\right]\_{p,q}}}^{1} \mathrm{td}\_{p,q} t \quad = \frac{\left[2\right]\_{p,q} - p^2}{\left[2\right]\_{p,q}^{3}},\tag{37}$$

$$\int\_{\frac{p}{\left[2\right]\_{p,q}}}^{1} (1-t)d\_{p,q}t \quad = \quad \frac{q\left[2\right]\_{p,q}^{2} + p^{2} - p - q}{\left[2\right]\_{p,q}^{3}}.\tag{38}$$

Making use of (34)–(38), gives us the required inequality (32). Hence, the proof is accomplished.

#### **6. Trapezoidal Type Inequalities through (***p***,** *q***)** *π***2 -Integral**

In this section, we give some new trapezoidal inequalities by using the (*p*, *q*)-derivative and integral.

To prove the main results of this section, we need the following crucial lemma.

**Lemma 3.** *Let* Π : *I* → R *be a differentiable function on I*◦*. If π*2*Dp*,*q*Π *is continuous and integrable on I, then we have the following identity:*

$$\frac{p\Pi(\pi\_2 + p\eta(\pi\_1, \pi\_2)) + q\Pi(\pi\_2)}{[2]\_{p,q}} - \frac{1}{p\eta(\pi\_2, \pi\_1)} \int\_{\pi\_2 + p\eta(\pi\_1, \pi\_2)}^{\pi\_2} \Pi(\varkappa) \, ^\pi \boldsymbol{d}\_{p,q} \varkappa$$

$$= -\frac{q\eta(\pi\_2, \pi\_1)}{[2]\_{p,q}} \int\_0^1 \left(1 - [2]\_{p,q}t\right) \, ^\pi \boldsymbol{2} \, \_{p,q} \Pi(t(\pi\_2 + \eta(\pi\_1, \pi\_2)) + (1 - t)\pi\_2) \, ^\pi \boldsymbol{d}\_{p,q} \mathbf{t}\_\prime \tag{39}$$
 $\pi\_2 = 0$ ,  $\pi\_2 = \pi\_2 = 1$

*where* 0 < *q* < *p* ≤ 1.

**Proof.** From (20) and the right side of (39), we obtain that

$$\begin{split} &\frac{q\eta\left(\pi\_{2},\pi\_{1}\right)}{[2]\_{p,q}}\int\_{0}^{1}\left(1-\left[2\right]\_{p,q}t\right)\,^{\pi\_{2}}D\_{p,q}\Pi\left(t\left(\pi\_{2}+\eta\left(\pi\_{1},\pi\_{2}\right)\right)+\left(1-t\right)\pi\_{2}\right)d\_{p,q}t \\ &=\ \ \frac{q\eta\left(\pi\_{2},\pi\_{1}\right)}{[2]\_{p,q}}\left[\frac{1}{\eta\left(\pi\_{2},\pi\_{1}\right)\left(p-q\right)}\int\_{0}^{1}\frac{-\Pi\left(qt\left(\pi\_{2}+\eta\left(\pi\_{1},\pi\_{2}\right)\right)+\left(1-qt\right)\pi\_{2}\right)}{t}\,d\_{p,q}t \\ &\quad -\frac{[2]\_{p,q}}{\eta\left(\pi\_{2},\pi\_{1}\right)\left(p-q\right)}\int\_{0}^{1}\left[\begin{matrix}\Pi\left(qt\left(\pi\_{2}+\eta\left(\pi\_{1},\pi\_{2}\right)\right)+\left(1-qt\right)\pi\_{2}\right) \\ -\Pi\left(qt\left(\pi\_{2}+\eta\left(\pi\_{1},\pi\_{2}\right)\right)+\left(1-qt\right)\pi\_{2}\right) \end{matrix}\,d\_{p,q}t .\end{split}$$

From (23) and (24), we have

$$\frac{q\eta(\pi\_{2\prime}\pi\_{1})}{[2]\_{p,q}}\int\_{0}^{1}\left(1-[2]\_{p,q}t\right)^{\pi\_{2}}D\_{p,q}\Pi(t(\pi\_{2}+\eta(\pi\_{1},\pi\_{2}))+(1-t)\pi\_{2})\,d\_{p,q}t$$

$$=\begin{array}{c} \begin{array}{c} q\eta\left(\pi\_{2},\pi\_{1}\right) \\ \hline \left[2\right]\_{p,q} \end{array} \left[ \begin{array}{c} \frac{\Pi(\pi\_{2})-\Pi(\pi\_{2}+\eta(\pi\mathfrak{r}\_{1},\pi\_{2}))}{\eta(\pi\_{2},\pi\_{1})} \\ -\frac{[2]\_{p,q}}{\eta(\pi\_{2},\pi\_{1})} \left\{\frac{1}{pq\eta(\pi\_{2},\pi\_{1})}\int\_{\pi\_{2}}^{\pi\_{2}}\eta\eta(\pi\_{1},\pi\_{2}) \end{array}\Pi(\varkappa)^{\frac{\pi\_{2}}{\pi\_{2}}}d\_{p,q}\varkappa-\frac{1}{q}\Pi(\pi\_{2}+\eta(\pi\_{1},\pi\_{2})) \right\} \end{array} \right] \end{array}$$

where the identity (39) is obtained and the proof is accomplished.

**Remark 15.** *We obtain Lemma 2 in [59] by letting p* = 1 *in Lemma 3.*

**Remark 16.** *If we adopt p* = 1 *in Lemma 3 and later we assume that η*(*π*2, *π*1) = −*η*(*π*1, *π*2) = *π*2 − *π*1*, then Lemma 3 becomes Lemma 1 in [41].*

**Theorem 8.** *Suppose that the assumptions of Lemma 3 hold. If π*2*Dp*,*q*Π *is a preinvex function over I, then we have the following new inequality:*

$$\begin{aligned} & \left| \frac{p\Pi(\pi\_2 + p\eta(\pi\_1, \pi\_2)) + q\Pi(\pi\_2)}{[2]\_{p,q}} - \frac{1}{p\eta(\pi\_2, \pi\_1)} \int\_{\pi\_2 + p\eta(\pi\_1, \pi\_2)}^{\pi\_2} \Pi(\varkappa) \, \prescript{\pi\_2}{}{}\_{p,q} \varkappa \right| \\ & \leq \left| \frac{q\eta(\pi\_2, \pi\_1)}{[2]\_{p,q}} \right| \left[ \left| \, \prescript{\pi\_2}{}{}\_{p,q}\Pi(\pi\_1) \right| A\_5(p,q) + \left| \, \prescript{\pi\_2}{}{}\_{p,q}\Pi(\pi\_2) \right| A\_6(p,q) \right], \end{aligned} \tag{40}$$

*where*

$$\begin{aligned} A\_5(p,q) &= \int\_0^1 t \left| \left(1 - \left[2\right]\_{p,q} t\right) \right| d\_{p,q} t, \\ A\_6(p,q) &= \int\_0^1 (1 - t) \left| \left(1 - \left[2\right]\_{p,q} t\right) \right| d\_{p,q} t. \end{aligned}$$

**Proof.** Taking the modulus in Lemma 3 and using the preinvexity of *π*2*Dp*,*q*Π , we have

$$\begin{split} & \left| \frac{p\Pi(\pi\_{2} + p\eta(\pi\_{1}, \pi\_{2})) + q\Pi(\pi\_{2})}{[2]\_{p,q}} - \frac{1}{p\eta(\pi\_{2}, \pi\_{1})} \int\_{\pi\_{2} + p\eta(\pi\_{1}, \pi\_{2})}^{\pi\_{2}} \Pi(\varkappa) \, \prescript{\pi\_{2}}{}{}\_{p,q} \varkappa \right| \\ & \leq \left| \frac{q\eta(\pi\_{2}, \pi\_{1})}{[2]\_{p,q}} \int\_{0}^{1} t \Big| \left(1 - [2]\_{p,q} t\right) \Big| \, \prescript{\pi\_{2}}{}{}\_{p,q} \Pi(\pi\_{1}) \right| \, d\_{p,q} t \\ & \quad + \int\_{0}^{1} (1 - t) \Big| \left(1 - [2]\_{p,q} t\right) \Big| \, \prescript{\pi\_{2}}{}{}{D\_{p,q}} \Pi(\pi\_{2}) \, \Big| \, d\_{p,q} t \\ & = \left. \frac{q\eta(\pi\_{2}, \pi\_{1})}{[2]\_{p,q}} \right| \left[ \, \prescript{\pi\_{2}}{}{}{D\_{p,q}} \Pi(\pi\_{1}) \right| A\_{5}(p,q) + \left| \, \prescript{\pi\_{2}}{}{}{D\_{p,q}} \Pi(\pi\_{2}) \right| A\_{6}(p,q) \Big]. \end{split} \tag{41}$$

Thus, the proof is completed.

**Remark 17.** *We obtain Theorem 7 in [59] by letting p* = 1 *in Theorem 8.*

**Remark 18.** *If we adopt p* = 1 *in Theorem 8 and later we assume that η*(*π*2, *π*1) = −*η*(*π*1, *π*2) = *π*2 − *π*1*, then Theorem 8 becomes Theorem 7 in [41].*

**Theorem 9.** *Suppose that the assumptions of Lemma 3 hold. If π*2*Dp*,*q*Π *r , r* ≥ 1 *is a preinvex function over I, then we have the following new inequality:*

$$\begin{split} & \left| \frac{p\Pi(\pi\_{2} + p\eta(\pi\_{1}, \pi\_{2})) + q\Pi(\pi\_{2})}{[2]\_{p,q}} - \frac{1}{p\eta(\pi\_{2}, \pi\_{1})} \int\_{\pi\_{2} + p\eta(\pi\_{1}, \pi\_{2})}^{\pi\_{2}} \Pi(\varkappa)^{\top 2} d\_{p,q} \varkappa \right| \\ & \leq \left. \frac{q\eta(\pi\_{2}, \pi\_{1})}{[2]\_{p,q}} \left( \int\_{0}^{1} \left| 1 - [2]\_{p,q}t \right| d\_{p,q}t \right)^{1 - \frac{1}{\tau}} \right. \\ & \left. \times \left[ \left| \begin{array}{c} \pi\_{2} \\ \end{array} D\_{p,q} \Pi(\pi\_{1}) \right|^{r} A\_{5}(p,q) + \left| \begin{array}{c} \pi\_{2} \\ \end{array} D\_{p,q} \Pi(\pi\_{2}) \right|^{r} A\_{6}(p,q) \right]^{\frac{1}{\tau}} \right. \tag{42} \end{split} \tag{42}$$

*where A*5(*p*, *q*) *and A*6(*p*, *q*) *are given in Theorem 8.*

**Proof.** Taking the modulus in Lemma 3 and applying the well-known power mean inequality for (*p*, *q*)-integrals and the preinvexity of *π*2*Dp*,*q*Π *r* , *r* ≥ 1, we obtain that

 *p*Π(*π*2 + *pη*(*π*1, *π*2)) + *q*Π(*π*2) [2]*p*,*q* − 1 *pη*(*π*2, *π*1) *π*2 *π*2+*pη*(*π*1,*π*2) Π(κ) *π*2 *dp*,*q*κ ≤ *qη*(*π*2, *π*1) [2]*p*,*q* - 1 0 1 − [2]*p*,*qt dp*,*qt* 1− 1 *r* × 1 0 1 − [2]*p*,*qt π*2*Dp*,*q*Π(*t*(*π*2 + *η*(*π*1, *π*2)) + (1 − *t*)*π*2) *r dp*,*qt* 1 *r* ≤ *qη*(*π*2, *π*1) [2]*p*,*q* - 1 0 1 − [2]*p*,*qt dp*,*qt* 1− 1 *r* 1 0 *t* 1 − [2]*p*,*qt π*2*Dp*,*q*Π(*π*1) *r dp*,*qt* + 1 0 (1 − *t*) 1 − [2]*p*,*qt π*2*Dp*,*q*Π(*π*2) *r dp*,*qt* 1 *r* = *qη*(*π*2, *π*1) [2]*p*,*q* - 1 0 1 − [2]*p*,*qt dp*,*qt* 1− 1 *r* × # *π*2*Dp*,*q*Π(*π*1) *r A*5(*p*, *q*) + *π*2*Dp*,*q*Π(*π*2) *r A*6(*p*, *q*) \$ 1 *r* . (43)

Thus, the proof is finished.

**Remark 19.** *We obtain Theorem 8 in [59] by letting p* = 1 *in Theorem 9.*

**Remark 20.** *If we adopt p* = 1 *in Theorem 9 and later we assume that η*(*π*2, *π*1) = −*η*(*π*1, *π*2) = *π*2 − *π*1*, then Theorem 9 becomes Theorem 4 in [41].*

**Theorem 10.** *Suppose that the assumptions of Lemma 3 hold. If π*2*Dp*,*q*Π *r , r* > 1 *is a preinvex function over I, then we have the following new inequality:*

$$\leq \frac{\left| p\Pi(\pi\_{2} + p\eta(\pi\_{1}, \pi\_{2})) + q\Pi(\pi\_{2}) - \frac{1}{p\eta(\pi\_{2}, \pi\_{1})} \int\_{\pi\_{2} + p\eta(\pi\_{1}, \pi\_{2})}^{\pi\_{2}} \Pi(\varkappa)^{\top 2} d\_{p, q} \varkappa \right|}{\left| \int\_{0}^{1} \right| - \left| 2 \right|\_{p, q} \left| \right|^{s} d\_{p, q} \Pi(\pi\_{1}) \left| \right|^{r} + \left( \left[ 2 \right]\_{p, q} - 1 \right) \left| \, \, \, ^{\pi\_{2}} D\_{p, q} \Pi(\pi\_{2}) \right|^{r} \right|^{\frac{1}{r}}},\tag{44}$$

*where s* + *r* = *sr*.

**Proof.** Taking the modulus in Lemma 3 and applying the well-known Hölder inequality for (*p*, *q*)-integrals and the preinvexity of *π*2*Dp*,*q*Π *r* , *r* > 1, we obtain that

$$\begin{split} & \left| \frac{p\Pi(\pi\_{2} + p\eta(\pi\_{1}, \pi\_{2})) + q\Pi(\pi\_{2})}{[2]\_{p,q}} - \frac{1}{p\eta\left(\pi\_{2}, \pi\_{1}\right)} \int\_{\pi\_{2} + p\eta\left(\pi\_{1}, \pi\_{2}\right)}^{\pi\_{2}} \Pi(\varkappa)^{\top \pi\_{2}} d\_{p,q} \varkappa \right| \\ & \leq \left| \begin{array}{c} \frac{q\eta\left(\pi\_{2}, \pi\_{1}\right)}{[2]\_{p,q}} \left( \int\_{0}^{1} \left| 1 - [2]\_{p,q}t \right|^{\frac{1}{s}} d\_{p,q}t \right)^{\frac{1}{s}} \\\\ & \times \left[ \int\_{0}^{1} \left| {}^{\pi\_{2}}D\_{p,q}\Pi(t(\pi\_{2} + \eta(\pi\_{1}, \pi\_{2})) + (1 - t)\pi\_{2}) \right|^{r} d\_{p,q}t \right]^{\frac{1}{r}} \end{split} \right| $$

$$\begin{split} \leq & \quad \frac{q\eta \left(\pi 2\_{2}\pi \pi\_{1}\right)}{\left[2\right]\_{p,q}} \left(\int\_{0}^{1} \left|1 - \left[2\right]\_{p,q}t\right|^{s} d\_{p,q}t\right)^{\frac{1}{s}} \\ \times & \left[\int\_{0}^{1} t \left|\,^{\pi\_{2}}D\_{p,q}\Pi(\pi\_{1})\right|^{r} d\_{p,q}t + \int\_{0}^{1} \left(1 - t\right) \left|\,^{\pi\_{2}}D\_{p,q}\Pi(\pi\_{2})\right|^{r} d\_{p,q}t\right]^{\frac{1}{r}}. \end{split} \tag{45}$$

We can calculate the integrals that occur on the right side of (45) as follows:

$$\int\_0^1 t \, d\_{p,q} t \quad = \quad \frac{1}{[2]\_{p,q}},\tag{46}$$

$$\int\_0^1 (1-t) \, d\_{p,q}t \quad = \quad \frac{[2]\_{p,q} - 1}{[2]\_{p,q}}.\tag{47}$$

Making use of (46) and (47) in (45), gives the desired result. Hence, the proof is completed.

**Remark 21.** *The left-right estimates of inequality (18) given in Theorem 4 that we left for the readers can be obtained by using the notions of* (*p*, *q*)*π*1 *-derivative and integral, as well as the techniques used in the previous two sections.*

### **7. Conclusions**

In this paper, we proved H-H type inequalities for preinvex functions using the (*p*, *q*)-calculus setup. For (*p*, *q*)-differentiable preinvex functions, we also proved some new midpoint-formula-type and trapezoid-formula-type inequalities. Furthermore, we demonstrated that the newly discovered inequalities are generalizations of the inequalities for convex functions in (*p*, *q*)-calculus. This study's conclusions can be used in symmetry. The results for symmetric functions can be reached by employing the notions of symmetric convex functions, which will be explored further in future work. It is an intriguing and novel problem, and future researchers will be able to obtain similar inequalities for coordinated preinvex functions in their studies.

**Author Contributions:** Conceptualization, I.B.S., M.A.A., G.M., S.K.N., J.S. and T.S.; methodology, I.B.S., M.A.A., G.M., S.K.N., J.S. and T.S.; formal analysis, I.B.S., M.A.A., G.M., S.K.N., J.S. and T.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by King Mongkut's University of Technology North Bangkok. Contract no. KMUTNB-63-KNOW-21.

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

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** We thank the referees for their valuable comments.

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

## **References**


**4. Conclusions**

From the analysis of the effect of Ti partial substitution by Mn, Fe, Co, or Cu on the NOx storage capacity (NSC) of the BaTi0.8B0.2O3 lean NOx trap (LNT) catalyst, the following conclusions have been obtained:

	- - (i) For the BTCuO\_2 catalyst, the tetragonal structure of the raw perovskite is distorted, a larger amount of oxygen vacancies is generated and Ba2TiO4 and BaCO3 appear as main minority segregated phases, but CuO is also detected.
	- - (ii) For Mn, Fe, and Co, the tetragonal structure changes to cubic, a lower amount of oxygen vacancies are formed and BaCO3 appears as segregated phase in the three catalysts. BTCoO\_2 presents also BaCoO3 and Co3O4 segregated phases due to a lower degree of Co insertion into the framework.

**Author Contributions:** Conceptualization, V.A.-F. and M.-J.I.-G.; Methodology, V.A.-F. and M.-J.I.-G.; Validation, V.A.-F, M.-S.S.-A., and M.-J.I.-G.; Formal analysis, C.A., V.T.-R., V.A.-F, M.-S.S.-A., and M.-J.I.-G.; Investigation, C.A., V.T.-R., V.A.-F.; Resources, M.-S.S.-A. and M.-J.I.-G.; Data curation, V.A.-F., M.-S.S.-A. and M.-J.I.-G.; Writing—original draft preparation, C.A., V.T.-R., and V.A.-F; Writing—review and editing, M.-S.S.-A. and M.-J.I.-G.; Visualization, C.A., V.A.-F., and M.-S.S.-A.; Supervision, V.A.-F. and M.-J.I.-G; Project administration, M.-J.I.-G.; Funding acquisition, M.-S.S.-A. and M.-J.I.-G.

**Funding:** This research was funded by Generalitat Valenciana (PROMETEO/2018/076 and Ph.D. grant ACIF 2017/221), Spanish Government (MINECO Project CTQ2015-64801-R) and EU (FEDER Founding).

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


*Edited by George Dekoulis*

This book, "Military Engineering", is a collection of reviewed and relevant research chapters, offering a comprehensive overview of the recent developments in the field of military engineering. The book comprises single chapters authored by various researchers and edited by an expert active in the physical sciences, engineering and technology research area. All chapters are complete in themselves but united under a common research study topic. This publication aims at providing a thorough overview of the latest research efforts by international authors on military engineering, and opening new possible research paths for further novel developments.

ISBN 978-1-78923-953-9

Military Engineering

Published in London, UK © 2020 IntechOpen © SuperheroTM / iStock


*3.1.2 Hall and pressure effects of Pr*2*Ba*4*Cu*7*O*15�*δ*

Furthermore, in **Figure 7**, we show the temperature dependences of the Hall coefficients *RH* for the as-sintered non-superconducting and 48-h-reduced superconducting samples of Pr2Ba4Cu7O15�*δ*. For comparison, the *RH* values of the assintered sample are taken from our previous work [33], which are similar, in magnitude and temperature dependence, to *RH* of Pr124 with a metallic CuO double-chain block. (**Figure 1a**) For the 48-h-reduced sample, the *RH* data exhibit negative values in the limited temperature range between 30 and 100 K, accompanied by electron doping due to the reduced heat treatment in a vacuum. Moreover, we estimate *RH* ¼ �1*:*1 � 10�3 cm3 /C at 30 K, which is in good agreement with the published data [33].

In **Figure 7b**, we show the temperature dependences of electrical resistivities of the 48-h-reduced superconducting sample under hydrostatic pressures up to 2.0 GPa. The application of external pressure on the 48-h-reduced sample suppressed the superconductivity with increasing the applied pressure. In the case

**Figure 7.**

*(a) Temperature dependences of Hall coefficients for the as-sintered non-superconducting and 48-h-reduced superconducting Pr247 compounds. For comparison, the as-sintered data are cited from our previous paper [34]. (b) Temperature dependences of electrical resistivities of the 48-h-reduced superconducting Pr247 under various pressures up to 2.0 GPa. (c) Magneto-resistance effect (up to 14 T) of the 48-h-reduced superconducting Pr247 for temperatures close to 30 K under a maximum pressure of 2.0GPa.*

of applied pressures above 0.8 GPa, the zero-resistance state vanished and the hightemperature metallic properties were transferred to the semi-conducting behaviors, accompanied by a rapid increase in *ρ*. The onset temperature of superconducting transition *T*c,on declined gradually from 26.5 K at ambient pressure through 24.1 K at 0.8 GPa down to 18.0 K at 1.6 GPa. However, the onset temperature was enhanced up to 30 K under a maximum pressure of 2.0 GPa.

The electronic phase diagram between normal and superconducting phases of a CuO double chain model has been clarified using the Tomonaga-Luttinger Liquid theory [32]. In the case of a shrinkage of the lattice spacing along *c*-axis between the two single chains of a CuO double-chain block, we expect the enhancement of both carrier hopping energies, *tpp* and *tdd*. Here, we define the hoping term between 2*pσ* orbitals at the nearest neighbor oxygen sites and that between 3*dx*2*y*2 orbitals at the nearest neighbor copper sites, as *tpp* and *tdd*, respectively (see **Figure 5b**). If we apply the external pressure on Pr247 including the CuO double chain block, it is theoretically predicted that the pressure induced enhancement of the hopping terms will result in a phase transition from the superconducting to normal phase. This theoretical prediction is qualitatively in agreement with the negative pressure effect on the superconducting phase observed in Pr247 [34].

We examined the magneto-resistance effect (up to 14 T) of the 48-h-reduced superconducting Pr247 for temperatures close to 30 K under a maximum pressure of 2.0 GPa, to establish a phase boundary between the superconducting and normal states. In **Figure 7c**, the MR data around 30 K tend to increase according to the upward covey behaviors at low fields. On the other hand, the MR around 40 K shows weak increases in the downward convey forms which is related to the model of slightly warped Fermi surfaces. (in details, see [27]) This finding indicates that the applied magnetic field destroyed the superconducting isolated regions and enlarged further the normal-state majority phase, resulting in the observed MR phenomena near 30 K. The re-entrant superconducting behavior observed at 2.0 GPa is an open question to be resolved in future through the structure analysis under applied pressures [35].

### **3.2 Double perovskite photocatalytic semiconductor**


**Stability Analysis and Stability Enhancement Based on Virtual Harmonic Resistance for Meshed DC Distributed Power Systems with Constant Power Loads**

#### **Huiyong Hu, Xiaoming Wang, Yonggang Peng \*, Yanghong Xia, Miao Yu and Wei Wei**

College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China<EMAIL_ADDRESS>(H.H<EMAIL_ADDRESS>(X.W<EMAIL_ADDRESS>(Y.X<EMAIL_ADDRESS>(M.Y<EMAIL_ADDRESS>(W.W.) **\*** Correspondence<EMAIL_ADDRESS>Tel.: +86-571-8795-2653

Academic Editor: Birgitte Bak-Jensen Received: 24 November 2016; Accepted: 2 January 2017; Published: 9 January 2017

**Abstract:** This paper addresses the stability issue of the meshed DC distributed power systems (DPS) with constant power loads (CPLs) and proposes a stability enhancement method based on virtual harmonic resistance. In previous researches, the network dynamics of the meshed DC DPS are often ignored, which affects the derivation of the equivalent system impendence. In addition, few of them have considered the meshed DC DPS including multiple sources with voltage-controlled converters and CPLs. To tackle the aforementioned challenge, this paper mainly makes the following efforts. The component connection method (CCM) is employed and expanded to derive the stability criterion of the meshed DC DPS with CPLs. This stability criterion can be simplified to relate only with the network node admittance matrix, the output impendences of the sources, and the input admittances of the CPLs. A virtual harmonic resistance through the second-order generalized integrator (SOGI) is added in the source with the voltage-controlled converter to lower the peak of the source output impendence, which can enhance the stability of the meshed DC DPS. The effectiveness of the proposed stability criterion and stability enhancement method are verified by nonlinear dynamic simulations.

**Keywords:** meshed DC distributed power systems; stability criterion; virtual harmonic resistance; stability enhancement

#### **1. Introduction**

With the increasing penetration of renewable energy generation into the modern electric girds, meshed DC distributed power systems (DPS) have been widely adopted [1–3]. The meshed DC DPS physically is composed of smaller power modules/subsystems. Usually, these smaller power modules/subsystems are designed only based on the stability requirement in its stand-alone operation, and thus it can operate well in the stand-alone application. However, the meshed DC DPS may become unstable due to the interaction among the modules/subsystems and the network [4]. Furthermore, the negative resistance characteristic of the constant power load (CPL) is an important unstable factor.

Many stability methods in previous researches are given to analyze the stability of the single-source-single-load system or the parallel-source-parallel-load system [4–6]. However, the network dynamics of the meshed DC DPS are often ignored, which affects the derivation of the equivalent system impendence. In addition, few of them have considered the meshed DC DPS including multiple sources with voltage-controlled converters and CPLs. A general approach for the stability analysis of the meshed DC DPS is to build its whole state-space model, and to identify the eigenvalues of the state matrix [7]. However, this method requires the detailed models of loads and network dynamics, and the formulation of the system matrices may be very complex. To overcome this

problem, the component connection method (CCM) is introduced for the stability analysis [8]. In the CCM, the dynamics of network and power modules/subsystems are modeled separately by a set of two vector-matrix equations, which is beneficial to reduce the computation burden of formulating the system transfer matrices [9]. For the ac power-electronics-based power system, the CCM is employed to derivate the closed-loop response transfer matrix of the overall power which can predict the instability of the power system [10], but it does not separate the source impedance and load admittance with the network node admittance matrix. Thus, it is difficult to obtain the general stability criterion intuitionally. In this paper, the CCM is also employed and expanded to derive the stability criterion for the meshed DC DPS with the CPLs. The stability criterion can be simplified to relate only with the network node admittance matrix, the output impendence of the source, and the input admittance of the CPLs which can be separately obtained by theoretical calculation or impedance analyzer measurement.

Many stability enhancement methods have been given in previous researches. An adaptive active capacitor converter (AACC) is introduced to effectively stabilize the cascaded system [11], but this method needs additional hardware devices. Another approach is to add a virtual adaptive parallel impedance [12] or adaptive series-virtual-impedance [13] in the input of the load converter to improve the stability in the cascaded DC/DC converter system. However, in the meshed DC DPS, it is not very effective due to the existence of an electromagnetic interference (EMI) filter for the CPL with a switching-mode power converter. The input admittance of the CPLs is mainly determined by the EMI filter, and the virtual impedance is difficult to be added by the load converter control part. Another way is to adjust the source output impedance. Modifying the source output impedance to be zero [14] and adding virtual impedances in the source output filters [15] are both feasible strategies by the source converter control part, but both of them will influence the low and high frequency characteristics of the source output impedance. To overcome the problem, this paper proposes an effective and achievable method, which is to add a virtual harmonic resistance through the second-order generalized integrator (SOGI) in the source converter control part to lower the peak of the source output impendence.

The main contributions of this paper are follows. Firstly, the stability criterion for the meshed DC DPS is derived, which has been simplified to relate only with the network node admittance matrix, the output impendence of the source, and the input admittance of the CPLs. It is easy to implement, as the impedance and the admittance can both be separately obtained by the theoretical calculation or impedance analyzer measurement. Secondly, an effective stability enhancement method by adding a virtual harmonic resistance is proposed, with which only the small range middle frequency characteristic of the source output impedance is modified, while the low and high frequency characteristics of the source output impedance are still unmodified. The results of the nonlinear dynamic simulations verify the effectiveness of the proposed stability criterion and stability enhancement method.

The remainder of this paper is organized as follows. The stability criterion for the meshed DC DPS is given in Section 2. The modeling of the source with voltage-controlled converters and the CPLs are derived, and the proposed stability enhancement method and the stability analysis for the meshed DC DPS are given in Section 3. Then, the nonlinear dynamic simulations are conducted in Section 4. At last, conclusions are drawn in Section 5.

#### **2. Stability Criterion for the Meshed DC DPS**

The general form of DC DPS is given in [4], but it ignores the network dynamics. In [4], any converter in a DC DPS can be classified as either a bus voltage controlled converter (BVCC) or a bus current controlled converter (BCCC). In this study, a four node meshed DC DPS as Figure 1 is considered as an example, which is composed by two sources with voltage-controlled converters and two CPLs. The sources with voltage-controlled converters are controlled with constant voltage, hence they can be treated as BVCCs. The CPLs are controlled to absorb constant power from the meshed DC DPS, hence they can be treated as BCCCs.

**Figure 1.** The block diagram of the component connection method (CCM) applied for the studied meshed DC distributed power systems (DPS).

Although all of the sources and CPLs are independently designed well, there will be complex interactions through the power cables in the meshed DC DPS. This consequently necessitates the use of CCM to analyze the stability of the meshed DC DPS.

The block diagram of the CCM applied for the studied meshed DC DPS is shown in Figure 1, where the CCM decomposes the overall DPS into four subsystems and the network. The two BVCCs are modeled by the Thevenin equivalent circuits [16], and the two BCCCs are modeled by the Norton equivalent circuits [17].

In Figure 1, *Zv*1 and *Zv*2 are the closed-loop output impedances of the sources, and *Yc*1 and *Yc*2 are the closed-loop input admittances of the CPLs. % *v*1 *v*2 *v*3 *v*4 &*T* is the node voltage column vector, and % *i*1 *i*2 *i*3 *i*4 &*T* is the node injection current column vector. % *icable*1 *icable*2 *icable*3 *icable*4 &*T* is the cable current column vector. *RL*3 and *RL*4 are the resistive loads connected to node 3 and node 4, respectively. % *Lcable*1 *Lcable*2 *Lcable*3 *Lcable*4 &*T* is the inductance column vector of cables, % *Rcable*1 *Rcable*2 *Rcable*3 *Rcable*4 &*T* is the parasitic resistance column vector of cables, and % *Ccable*1 *Ccable*2 *Ccable*3 *Ccable*4 &*T* is the capacitance column vector of cables. % *Gv*1 *Gv*2 *Gc*1 *Gc*2 &*T* is the voltage reference-to-output transfer function column vector of the source and CPLs. % *vv*1*ref vv*2*ref vc*1*ref vc*2*ref* &*T* is the reference voltage column vector of the source and CPLs, respectively.

Generally, the source is designed to be a voltage source that is stable when unloaded, and the CPL is designed to be stable when supplied by an ideal voltage source. That is, the unterminated behaviors of inverters % *Gv*1 *Gv*2 *Gc*1 *Gc*2 &*T* are stable, and % *Zv*1 *Zv*2 *Yc*1 *Yc*2 &*T* are stable, which means that there are no right-half plane poles.

To facilitate the formulation of the nodal admittance matrix, the Thevenin circuits of the BVCCs are converted to Norton circuits [10]. Then, the studied meshed DC DPS is represented as Figure 2.

**Figure 2.** The equivalent Norton circuit of the studied meshed DC DPS.

Applying the node voltage equation to the external dashed box part in Figure 2, it can be derived as

$$
\begin{bmatrix}
\begin{bmatrix}
Z\_{v1}^{-1}G\_{v1}\upsilon\_{v1ref} \\
Z\_{v2}^{-1}G\_{v2}\upsilon\_{v2ref} \\
G\_{\ell1}\upsilon\_{\ell1ref} \\
G\_{\ell2}\upsilon\_{\ell2ref}
\end{bmatrix} =
\begin{pmatrix}
\begin{bmatrix}
Z\_{v1}^{-1} \\
Z\_{v2}^{-1} \\
\upsilon\_{\ell1} \\
\upsilon\_{\ell2}
\end{bmatrix} + \chi\_{\rm sys} \\
\begin{bmatrix}
\upsilon\_{\ell1} & \upsilon\_{\ell2}
\end{bmatrix}
\end{pmatrix} + \chi\_{\rm sys}
\begin{bmatrix}
\upsilon\_{1} \\
\upsilon\_{2} \\
\upsilon\_{3}
\end{bmatrix}
\end{bmatrix}
\tag{1}
$$

where *Ysys* is the network nodal admittance matrix as

$$\mathbf{Y}\_{sp} = \begin{bmatrix} \mathbb{C}\_{\text{idid}:0} + \mathbb{C}\_{\text{idid}:0} + \mathbf{Y}\_{\text{idid}} + \mathbf{Y}\_{\text{idid}:2} & \mathbf{0} & -\mathbf{Y}\_{\text{idid}:2} & -\mathbf{Y}\_{\text{idid}:1} \\ \mathbf{0} & \mathbb{C}\_{\text{idid}:0} + \mathbb{C}\_{\text{idid}:0} + \mathbf{Y}\_{\text{idid}} + \mathbf{Y}\_{\text{idid}} & -\mathbf{Y}\_{\text{idid}} & -\mathbf{Y}\_{\text{idid}:1} & \mathbf{0} \\ -\mathbf{Y}\_{\text{idid}:2} & -\mathbf{Y}\_{\text{idid}:0} & \mathbf{Y}\_{\text{id}} + \mathbb{C}\_{\text{idid}:0} + \mathbb{C}\_{\text{idid}:0} + \mathbf{Y}\_{\text{idid}:2} + \mathbf{Y}\_{\text{idid}:1} & \mathbf{0} & \mathbf{0} \\ -\mathbf{Y}\_{\text{idid}:1} & -\mathbf{Y}\_{\text{idid}:1} & \mathbf{0} & \mathbf{Y}\_{\text{idid}:1} + \mathbb{C}\_{\text{idid}:0} + \mathbf{Y}\_{\text{idid}:1} & \mathbf{0} & \mathbf{Y}\_{\text{idid}:1} \\ \end{bmatrix} \tag{2}$$

where *YL*3 and *YL*4 are the admittances for loads connected to node 3 and node 4, respectively. *Ycable*1~*Ycable*4 are the admittances for cables and *Ycablei* = 1/(*Rcablei* + *Lcableis*), *i* = 1∼4 .

Then, the node voltage can be derived from (1) as follows:

$$
\begin{aligned}
\begin{bmatrix} v\_1 \\ v\_2 \\ v\_3 \\ v\_4 \end{bmatrix} &= \left( \begin{bmatrix} Z\_{v1}^{-1} \\ & Z\_{v2}^{-1} \\ & & Y\_{c1} \\ & & & Y\_{c2} \end{bmatrix} + Y\_{sys} \right)^{-1} \begin{bmatrix} Z\_{v1}^{-1} \\ & Z\_{v2}^{-1} \\ & & 1 \\ & & & 1 \end{bmatrix} \begin{bmatrix} G\_{v1}v\_{v1ref} \\ G\_{v2}v\_{v2ref} \\ G\_{v1}v\_{v1ref} \\ G\_{v2}v\_{v2ref} \end{bmatrix} \tag{3} \\\\ v &= \left( \begin{bmatrix} Z\_v^{-1} \\ & Y\_c \\ & & Y\_c \end{bmatrix} + Y\_{sys} \right)^{-1} \begin{bmatrix} Z\_v^{-1} \\ & I\_c \\ & & I\_c \end{bmatrix} \begin{bmatrix} G\_v \\ & G\_v \\ & G\_{vc} \end{bmatrix} \begin{bmatrix} v\_{vref} \\ v\_{vref} \\ v\_{cref} \\ v\_{cref} \end{bmatrix} \end{aligned} \tag{4}
$$

where *Zv* is the set of the closed-loop output impedances of the sources, and *Yc* is the set of the closed-loop input admittances of the CPLs. *v* is the set of node voltages, and *i* is the set of the node injection currents. *Gv* is the the set of voltage reference-to-output transfer functions of the source, *Gc* is the the set of voltage reference to output transfer functions of the CPLs, *vvre f* is the set of voltage references of the source, and *vcre f* is the set of voltage references of the CPLs. *Ic* is a unit diagonal

matrix whose dimension is equal to *Yc*. *Iv* is a unit diagonal matrix whose dimension is equal to *Zv*. And they satisfy the following relations:

$$\mathbf{Z}\_{v} = \begin{bmatrix} \mathbf{Z}\_{v1} \\ & \mathbf{Z}\_{v2} \end{bmatrix}, \mathbf{Y}\_{c} = \begin{bmatrix} \mathbf{Y}\_{c1} \\ & \mathbf{Y}\_{c2} \end{bmatrix}, \mathbf{G}\_{v} = \begin{bmatrix} \mathbf{G}\_{v1} \\ & \mathbf{G}\_{v2} \end{bmatrix}, \mathbf{G}\_{c} = \begin{bmatrix} \mathbf{G}\_{c1} \\ & \mathbf{G}\_{c2} \end{bmatrix}, \mathbf{v}\_{\text{var}f} = \begin{bmatrix} \mathbf{v}\_{c1\text{rf}f} \\ \mathbf{v}\_{c2\text{rf}f} \end{bmatrix}, \mathbf{v}\_{\text{var}f} = \begin{bmatrix} \mathbf{v}\_{c1\text{rf}f} \\ \mathbf{v}\_{c2\text{rf}f} \end{bmatrix} \tag{5}$$

$$\mathbf{v}\_{\text{var}} = \begin{bmatrix} \mathbf{v}\_{c1\text{rf}f} \\ \mathbf{v}\_{c2\text{rf}f} \end{bmatrix}$$

$$v = \begin{bmatrix} v\_1 & v\_2 & v\_3 & v\_4 \end{bmatrix}^T,\\ i = \begin{bmatrix} i\_1 & i\_2 & i\_3 & i\_4 \end{bmatrix}^T \tag{6}$$

Applying the node voltage equation to the internal dashed box part in Figure 2, *i* can be derived as:

$$\mathbf{M} = \mathbf{Y}\_{\text{sys}} \mathbf{v} = \mathbf{Y}\_{\text{sys}} \left( \begin{bmatrix} I\_{\text{v}} \\ & Y\_{\text{c}} \end{bmatrix} + \begin{bmatrix} Z\_{\text{v}} \\ & I\_{\text{c}} \end{bmatrix} \mathbf{Y}\_{\text{sys}} \right)^{-1} \begin{bmatrix} \mathbf{G}\_{\text{v}} \\ & \mathbf{G}\_{\text{c}} \end{bmatrix} \begin{bmatrix} v\_{\text{v}ref} \\ v\_{\text{c}ref} \end{bmatrix} \tag{7}$$

In the above equation, *Ysys* is the network nodal admittance matrix for a real physical system, which means *Ysys* is stable and has no right-half plane poles. *Gv* and *Gc* are both stable as in the above analysis. Therefore, the stability criterion of the meshed DC DPS *Tm* is

$$T\_m = \left( \left[ \begin{array}{cc} I\_v \\ & Y\_c \end{array} \right] + \left[ \begin{array}{cc} Z\_v \\ & I\_c \end{array} \right] Y\_{\text{sys}} \right)^{-1} \tag{8}$$

If the stability criterion *Tm* has right-half plane poles, the meshed DC DPS will be unstable. Through the analysis of the stability criterion *Tm*, it can be found that the stability of the meshed DC DPS is only related to the network node admittance matrix *Ysys*, the output impendence of the sources *Zv*, and the input admittance of the CPLs *Yc*, which can be separately obtained by theoretical calculation or impedance analyzer measurement.

#### **3. Modeling, Stability Enhancement Method and Stability Analysis**

#### *3.1. Modeling of the CPL*

The modeling of the CPL connected to node 3 is similar to the CPL connected to node 4. Without loss of generality, taking the CPL connected to node 4 as an example. The CPL usually exploits a switching-mode power converter as the interface with the node, and the compliance with EMI standards usually requires the insertion of an EMI filter between the switching-mode power converter and the node [18–20]. The close-loop circuit of the CPL with an EMI filter is shown as Figure 3.

**Figure 3.** The close-loop circuit of the constant power load (CPL) with an electromagnetic interference (EMI) filter.

Applying switch period average, the small-signal model of the CPL with EMI filter can be achieved as Figure 4.

**Figure 4.** The small-signal model of the CPL with an EMI filter.

The variables showing in capital letters represent the steady-state values of duty cycle, inductor current, capacitor voltage, and output current, and the variables with " ˆ " indicate the small-signal disturbance.

It can be derived from Figure 4 that the small-signal disturbance of the output current ˆ*i*4 can be expressed as:

$$
\hat{\mathfrak{u}}\_4 = -\mathcal{Y}\_{\mathfrak{c2}}\mathfrak{d}\_4 + \mathcal{G}\_{\mathfrak{c2}}\mathfrak{d}\_{\mathfrak{c2}ref} \tag{9}
$$

$$Y\_{c2} = \frac{a\_4s^4 + a\_3s^3 + a\_2s^2 + a\_1s^1 + a\_0}{\Delta\_{c2}} \tag{10}$$

$$G\_{c2} = -\frac{c\_3s^3 + c\_2s^2 + c\_1s^1 + c\_0}{\Delta\_{c2}}\tag{11}$$

$$
\Delta\_{c2} = b\_5 s^5 + b\_4 s^4 + b\_3 s^3 + b\_2 s^2 + b\_1 s^1 + b\_0 \tag{12}
$$

where *Gc*2*v* is the voltage controller, *Yc*2 is the closed-loop input admittance of the CPL, and *Gc*2 is the voltage reference to output transfer function of the CPL. *a*0 ∼*a*4 , *b*0 ∼*b*5, and *c*0 ∼*c*3 are given in Appendix A.

#### *3.2. Modeling of the Source with Voltage-Controlled Converters*

The modeling of the source with voltage-controlled converters connected to node 1 is similar to the source connected to node 2. Without loss of generality, taking the source connected to node 1 as an example, the close-loop circuit of the source with voltage-controlled converters is shown as Figure 5.

**Figure 5.** The close-loop circuit of the source with voltage-controlled converters (where *Gv*1*v* and *Gv*1*i* are the voltage and current controllers).

Figure 6 shows the small-signal block diagram of the multi-loop control system of the source with voltage-controlled converters.

**Figure 6.** The small-signal block diagram of the multi-loop control system of the source.

Thus, the dynamic behavior of the close-loop system can be given by

$$
\psi\_1 = -Z\_{v1}\hat{\imath}\_1 + G\_{v1}\vartheta\_{v1ref} \tag{13}
$$

where

$$Z\_{v1} = \frac{G\_{v1i}K\_{v1pwvu}V\_1 + L\_1s + r\_1}{L\_1\mathbb{C}\_1s^2 + (\mathbb{C}\_1\mathbb{G}\_{v1i}K\_{v1pwv}V\_1 + r\_1\mathbb{C}\_1)s + G\_{v1i}G\_{v1v}K\_{v1pwv}V\_1 + 1} \tag{14}$$

$$\mathbf{G}\_{v1} = \frac{\mathbf{G}\_{v1i}\mathbf{G}\_{v1v}\mathbf{K}\_{v1pwm}V\_1}{L\_1\mathbf{C}\_1\mathbf{s}^2 + (\mathbf{C}\_1\mathbf{G}\_{v1i}\mathbf{K}\_{v1pwm}V\_1 + r\_1\mathbf{C}\_1)\mathbf{s} + \mathbf{G}\_{v1i}\mathbf{G}\_{v1v}\mathbf{K}\_{v1pwm}V\_1 + 1} \tag{15}$$

where *Zv*1 is the closed-loop output impedance of the source with voltage-controlled converters, and *Gv*1 is the voltage reference-to-output transfer function of the source with voltage-controlled converters.

#### *3.3. Stability Enhancement Method*

The source output impedance has a peak at the resonant frequency. Based on the Nyquist criterion, instability between the source and the network may occur if the network input impedance becomes lower than the source output impedance at this frequency. Therefore, to ensure the system stability and to minimize the potential for inadvertent interactions with the source, it is important to lower the peak of the source output impedance. For this purpose, this paper proposes a method by adding a virtual harmonic resistance through the SOGI to be in parallel with the capacitor.

The transfer function *Gf*(*s*) of SOGI can be expressed as

$$G\_f(\mathbf{s}) = \frac{\mathbf{s}^2 + \omega^2}{\mathbf{s}^2 + k\omega\mathbf{s} + \omega^2} \tag{16}$$

where *ω* is the resonant frequency, and *k* is the frequency coefficient. The frequency-domain characteristic of *Gf*(*s*) is shown as Figure 7. Around the resonant frequency, the amplitude of *Gf*(*s*) is very small, and the amplitude of *Gf*(*s*) in other frequency ranges is 0 dB.

**Figure 7.** The frequency-domain characteristic of *Gf*(*s*).

Figure 8 shows the small-signal control block of the source 1. If a virtual harmonic resistance *Rvh* is required to be added in parallel with the capacitor, it is intuitively to introduce *Rvh* to the block as a dot-dashed part in Figure 8, which is the basic idea of this control method. However, this method cannot be realized by control directly. Therefore, *Rvh* is moved to the output of *Gv*1*v*(*s*) as the dashed part in Figure 8.

**Figure 8.** Adding a virtual harmonic resistance through the second-order generalized integrator (SOGI).

*GR*(*s*) is expressed as

$$G\_R(s) = \frac{L\_1s + r\_1 + G\_{v1i}K\_{v1puvu}V\_1}{G\_{v1i}K\_{v1puvu}V\_1R\_{vlu}}.\tag{17}$$

With the parallel virtual harmonic resistance *Rvh* = 1 Ω, the frequency-domain characteristic of the source output impedance is shown as Figure 9.

**Figure 9.** The source output impedance with/without parallel virtual harmonic resistance *Rvh*.

As can be seen, the peak of the source output impedance has been lower, and only the small range middle frequency characteristic of the source output impedance is modified, while the low and high frequency characteristics of the source output impedance are still unmodified.

#### *3.4. Stability Analysis*

All the main electrical parameters of the studied DC meshed DPS as shown in Figure 1 are listed in Appendix B.

The closed-loop output impedances of the sources *Zv* can be represented by using impedance calculated in Equations (5) and (14), the closed-loop input admittances of the CPLs *Yc* can be represented by using admittance calculated in Equations (5) and (10), and the network nodal admittance matrix *Ysys* can be represented by using admittance calculated in Equation (2). Then, the the stability of the meshed DC DPS can be analyzed with the stability criterion *Tm* expressed in Equation (8). If the stability criterion *Tm* has right-half plane poles, the meshed DC DPS will be unstable.

At first, it is necessry to compare the proposed stability criterion *Tm* in this paper and the previous stability criterion *Tm*2 given in [4] as follows:

$$T\_{m2} = \frac{1}{1 + (Z\_{v1} // Z\_{v2})(Y\_{c1} + Y\_{c2} + Y\_{L3} + Y\_{L4})}.\tag{18}$$

The dominant poles of *Tm* ans *Tm*2 are displayed as Figure 10, in which the cables are long *Rcablei* = 0.05 Ω , *Lcablei* = 0.5 mH, *Ccablei* = 100 μF, and *i* = 1∼4. As shown in Figure 10, all the dominant poles of *Tm*2 are both in the left-half plane, which indicates that the meshed DC DPS is stable. However, there are two right-half plane poles 0.15 ± 94.5*i* in the dominant poles of *Tm*, which indicates the meshed DC DPS is unstable and have an oscillation with approximately period 0.066 s. The results of the proposed stability criterion *Tm* and the previous stability criterion *Tm*2 are not consistent, and the results of proposed stability criterion *Tm* will be confirmed to be right with the nonlinear dynamic simulations in Section 4. Moreover, many poles related to the network dynamics are also lost in *Tm*2 compared to *Tm*.

**Figure 10.** The dominant poles of *Tm* ans *Tm*2 when the cables are long.

With the electrical parameters listed in Table B1, when the power of resistive load 4 *PL*4 is 50 kW, which means *RL*4 = 5 Ω, the dominant poles of *Tm* are displayed as Figure 11. When the power of resistive load 4 *PL*4 reduces from 50 kW to 10 kW, which means *RL*4 = 25 Ω, the dominant poles of *Tm* are displayed as Figure 12. Furthermore, when the power of resistive load 4 *PL*4 is 10 kW and the absorbed power of CPL2 *Pc*2 reduces from 100 kW to 50 kW, which means *RL*4 = 25 Ω and *Rc*2 = 1.25 Ω, and the dominant poles of *Tm* are displayed as Figure 13.

**Figure 11.** The dominant poles of *Tm* when *PL*4 = 50 kW, *Pc*2 = 100 kW.

**Figure 12.** The dominant poles of *Tm* when *PL*4 = 10 kW, *Pc*2 = 100 kW.

**Figure 13.** The dominant poles of *Tm* when *PL*4 = 10 kW, *Pc*2 = 50 kW.

As can be seen from Figure 11, all of the poles of *Tm* are in the left half plane, which indicates that the meshed DC DPS is stable. But there are two right half plane poles 0.48 ± 97*i* in the Figure 12, which indicates the meshed DC DPS is unstable and have an oscillation with approximately period 0.0647 s. Furthermore, with the power reducation of CPL2, the meshed DC DPS becomes stable as shown in Figure 13.

Applying the proposed stability enhancement method in the sources connected with node 1 and node 2 of the meshed DC DPS when *PL*4 is 10 kW and *Pc*2 is 100 kW, the dominant poles of *Tm* are displayed as Figure 14. There are no right half plane poles, which means that the meshed DC DPS is stable. Compared to the poles in Figure 12, the stability of the meshed DC DPS is enhanced by the proposed stability enhancement method.

**Figure 14.** The dominant poles of *Tm* with the proposed stability enhancement method.

#### **4. Nonlinear Dynamic Simulations**

To validate the effectiveness of the proposed stability criterion and stability enhancement method, the meshed DC DPS in Figure 1 is built in the nonlinear time-domain simulation by using Matlab/Simulink. The electrical and controller parameters are all given in Appendix B.

To compare the proposed stability criterion *Tm* and the previous stability criterion *Tm*2, the cables are chosen to be long cables with parameters *Rcablei* = 0.05 Ω , *Lcablei* = 0.5 mH, *Ccablei* = 100 μF, *i* = 1∼4, the nonlinear dynamic simulation results of the node 1 voltage *v*1(*t*) and the cable 1 current *icable*1(*t*) are shown in Figure 15. As can be seen in Figure 15, before 2.5 s, the meshed DC DPS gradually becomes to a stable state with *PL*4 = 50 kW, and once the power of resistive load 4 *PL*4 reduces to 15 kW at 2.5 s, the meshed DC DPS becomes to an oscillation state. The oscillation amplitude gradually grows, and the oscillation period is about 0.066 s, which satisfy the theoretical analysis in Figure 10. The results verify the effectiveness of the proposed stability criterion *Tm* compared to the previous stability criterion *Tm*2 which does not take the network dynamics into account.

**Figure 15.** Comparison between *Tm* and *Tm*2.

To verify the effectiveness of the proposed stability criterion *Tm* with power changes of the resistive load and CPL, and the proposed stability enhancement method, the nonlinear dynamic simulation results are shown as the following Figures 16–18.

As can be seen in Figure 16, before 1.5 s, the meshed DC DPS gradually becomes to stable state with *PL*4 = 50 k*W* and *Pc*2 = 100 k*W*, which confirms the theoretical analysis in Figure 11. Then, the power of resistive load 4 *PL*4 reduces to 10 kW at 1.5 s, the meshed DC DPS becomes to the oscillation state with about period 0.065 s, which confirms the theoretical analysis in Figure 12. The power of CPL2 reduces from 100 kW to 50 kW at 2 s, and the meshed DC DPS becomes to a stable state again, which confirms the theoretical analysis in Figure 13.

**Figure 16.** The node 1 voltage *v*1(*t*) with power changes of the resistive load and CPL.

**Figure 17.** The node 1 voltage *v*1 without/with the proposed stability enhancement method.

**Figure 18.** The currents of cable 1 *icable*1 and cable 4 *icable*4.

As can be seen in Figure 17, when the power of CPL2 increases from 50 kW to 100 kW at 3.5 s, the meshed DC DPS becomes to the oscillation state again between 3.5–4.5 s. Then, the vitural harmonic resistances are added in the controllers of source 1 and source 2 at 4.5 s. With the vitural harmonic resistance, the meshed DC DPS becomes to the stable state again, and the node 1 voltage *v*1 converges again to the steady value 500 V. The simulation results between 4.5–5.5 s confirm the theoretical analysis results in Figure 14. The power of resistive load 4 *PL*4 increases to 50 kW at 5.5 s, and the meshed DC DPS is maintained at a steady state.

The currents of cable 1 *icable*1 and cable 4 *icable*4 are displayed as Figure 18. The steady state and unstable state are similar to the node 1 voltage *v*1 as shown in Figures 16 and 17. It is worth noting that the steady state current values in 0–1.5 s and 5.5–6 s are the same, which means that with or without the proposed stability enhancement method, the current distributions in the meshed DC DPS are not affected, which verifies that the low frequency characteristic of the source output impedance has not been modified.

#### **5. Conclusions**

This paper discusses the modeling and stability analysis of the meshed DC DPS including multiple sources with voltage-controlled converters and CPLs, and proposes a stability enhancement method. The stability criterion is found to relate only with the network node admittance matrix, the output impendence of the source, and the input admittance of the CPLs, and the three parts can be obtained separately. Virtual harmonic resistance through the second-order generalized integrator (SOGI) for the source proposed in this paper is a decoupling design which is only based on the characteristic of the source output impedance. Nonlinear dynamic time-domain simulation results verify the effectiveness of the proposed stability criterion and stability enhancement method.

Furthermore, the current control sources or loads can be treated as BCCCs, and the input admittances of current control sources or loads are related to the stability analysis. Multiple sources with droop control can be treated as BVCCs, and the output impendence of the droop control sources are related to the stability analysis. So the proposed stability criterion can also deal with the current control sources or loads and multiple sources with droop control. Related analyses are in the future researches.

**Acknowledgments:** The research leading to these results has received funding from the Nation "863" project of China (2014AA052001), the National Natural Science Foundation of China (51377142), the Zhejiang Provincial Natural Science Foundation of China (LY16E070002), and the Key Research and Development Project of Zhejiang Province (2016C01G2010726).

**Author Contributions:** All authors contributed equally in the presented research, and all authors approved the final manuscript.

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

#### **Appendix A**

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ *a*0 = *D*2 − *Gc*2*v Ic*2*Kc*2*pwmRc*2*D a*1 = −*Rc*2*Rd f* 2*Cd f* 2*Gc*2*v Ic*2*Kc*2*pwmD* + (*Cd f* 2 + *Cf* 2)(*Gc*2*vKc*2*pwmRc*2*Vf* 2 + *Rc*2 + *rc*2)+(*Rc*2*Cc*2 + *Rd f* 2*Cd f* 2)*D*2 *a*2 = *Rc*2*Rd f* 2*Cd f* 2(*Cf* 2*Gc*2*vKc*2*pwmVf* 2 + *Cc*2*D*2) + *Cd f* 2*Cf* 2*Rd f* 2(*Rc*2 + *rc*2)+(*Lc*2 + *Cc*2*Rc*2*rc*2)(*Cd f* 2 + *Cf* 2) *a*3 = (*rc*2*Rc*2*Cc*2 + *Lc*2)*Rd f* 2*Cd f* 2*Cf* 2 + *Cc*2*Lc*2*Rc*2(*Cf* 2 + *Cd f* 2 *a*4 = *Rc*2*Cc*2*Rd f* 2*Cd f* 2*Lc*2*Cf* 2 (A1) *b*0 = *Gc*2*vKc*2*pwmRc*2*Vf* 2 + *Rc*2 + *rc*2 *b*1 = (*Rc*2*Rd f* 2*Cd f* 2*Vf* 2 − *Rc*2*D Ic*2*Lf* 2)*Gc*2*vKc*2*pwm* + *Rc*2*Cc*2*rc*2 + *Rd f* 2*Cd f* 2(*Rc*2 + *rc*2) + *D*2*Lf* 2 + *Lc*2 *b*2 = −*Rc*2*Rd f* 2*Cd f* 2*D Ic*2*Gc*2*vKc*2*pwmLf* 2 + (*Cd f* 2 + *Cf* 2)(*Rc*2*Vf* 2*Gc*2*vKc*2*pwmLf* 2 + *Rc*2*Lf* 2 + *rc*2*Lf* 2)+(*Rc*2*Cc*2 + *Rd f* 2*Cd f* 2)(*Lf* 2*D*2 + *Lc*2) + *Rc*2*Cc*2*Rd f* 2*Cd f* 2*rc*2 *b*3 = *Rc*2*Rd f* 2*Cd f* 2*Cf* 2*Gc*2*vKc*2*pwmLf* 2*Vf* 2 + (*D*2*Lf* 2 + *Lc*2)*Rc*2*Cc*2*Rd f* 2*Cd f* 2 + *Cd f* 2*Cf* 2*Lf* 2*Rd f* 2(*Rc*2 + *rc*2)+(*Cd f* 2 + *Cf* 2)(*Cc*2*Lf* 2*Rc*2*rc*2 + *Lc*2*Lf* 2) *b*4 = *Cd f* 2*Cf* 2*Lf* 2*Rd f* 2(*Rc*2*rc*2*Cc*2 + *Lc*2 + *Cd f* 2 + *Cf* 2)*Cc*2*Lc*2*Lf* 2*Rc*2 *b*5 = *Cc*2*Cd f* 2*Cf* 2*Lc*2*Lf* 2*Rc*2*Rd f* 2 (A2) ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ *c*0 = *Gc*2*vKc*2*pwmVf* 2*D* + *Gc*2*v Ic*2*Kc*2*pwm*(*Rc*2 + *rc*2) *c*1 = (*Cc*2*Rc*2 + *Cd f* 2*Rd f* 2)*Gc*2*vKc*2*pwmVf* 2*D* + *Lc*2 + *Rc*2*Rd f* 2*Cd f* 2 + *rc*2*Rd f* 2*Cd f* 2 + *rc*2*Rc*2*Cc*2)*Gc*2*vKc*2*pwm Ic*2 *c*2 = *Rc*2*Cc*2*Rd f* 2*Cd f* 2*Gc*2*vKc*2*pwmVf* 2*D* + *Rc*2*Cc*2*Lc*2 + *Rd f* 2*Cd f* 2*Lc*2 + *Rc*2*Cc*2*Rd f* 2*Cd f* 2*rc*2)*Gc*2*v Ic*2*Kc*2*pwm* (A3)

```
c3 = Rc2Cc2Rd f 2Cd f 2Lc2Gc2v Ic2Kc2pwm
```
#### **Appendix B**

This Appendix lists all the main electrical parameters of the studied DC meshed DPS as shown in Figure 1.

**Table B1.** Main electrical parameters of the studied DC meshed distributed power systems (DPS).

#### **References**

© 2017 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/).

MDPI St. Alban-Anlage 66 4052 Basel Switzerland Tel. +41 61 683 77 34 Fax +41 61 302 89 18 www.mdpi.com

*Energies* Editorial Office E-mail<EMAIL_ADDRESS>www.mdpi.com/journal/energies

MDPI St. Alban-Anlage 66 4052 Basel Switzerland

Tel: +41 61 683 77 34 Fax: +41 61 302 89 18


**6.2. Exception of equipment or operating parameters**

The secondary cooling process is the most important procedure in continuous casting. The temperature field of the slab was checked with the data provided by the producer of the caster. A good agreement was found, indicating that the caster worked well in the secondary cooling process. The monitoring records obtained in a controlled room also reveal that the caster worked well. However, the width of the produced slabs exhibited obvious broadening during the process. It is thus concluded that slab broadening is independent of exceptions of equipment or operating parameters.


**7. Organs at risk delineation**

Organs at risk including heart, esophagus, and spinal cord will be contoured using soft tis‐ sue window. The heart includes the whole structure within pericardial sac starting from the pulmonary artery to the apex. All layers of esophagus will be included and contoured from the cricoid cartilage to the esophagogastric junction. Spinal cord will be contoured at least 10 cm above and below PTV. For tumors over upper chest, the ipsilateral brachial plexus should also be contoured. Both left and right lungs are also contoured and then used to form a new structure called lung minus PTV after subtraction.

For SBRT, the tracheal and proximal bronchial tree should be contoured as well. Trachea will start from the level of cricoid cartilage to 2 cm above the carina, where it then con‐ tinuous with the proximal bronchial tree (PBT, **Figure 13**) including the distal 2 cm trachea, main carina, bilateral main bronchi, bilateral upper lobe bronchi, lingular bron‐ chus, intermedia bronchus, right middle lobe bronchus, and bilateral lower lobe bronchi. A 2 cm margin applied around the PTB will then be used to form a PRV (planning organ at risk volume).

**Figure 13.** Definition of proximal bronchial tree.


**M. Awais 1, Zahir Shah 2,\*, N. Perveen 1, Aamir Ali 1, Poom Kumam 3,4,5,\*, Habib ur Rehman 3 and Phatiphat Thounthong 6**

Received: 27 November 2019; Accepted: 12 February 2020; Published: 19 February 2020

**Abstract:** Present theoretical investigation is a mathematical illustration of an application to endoscopy by incorporating hybrid nanoparticles and an induced magnetic field with a rheological fluid model for more realistic results. Rheological fluid behavior is characterized by the Ostwald-de-Waele power-law model. A hybrid nanofluid mechanism is considered comprising platelet-shaped nanoparticles since nanoparticles are potential drug transportation tools in biomedical applications. Moreover, ciliary activity is encountered regarding their extensive applications in performing complex functions along with buoyancy effects. An endoscope is inserted inside a ciliated tube and peristalsis occurred due to ciliary activity in the gap between tube and endoscope. A non-Newtonian model is developed by mathematical formulation which is tackled analytically using homotopy analysis. The outcomes are interpreted graphically along with the pressure rise and streamlining configuration for the case of negligible inertial forces and long wavelength. A three-dimensional graphical interpretation of axial velocity is studied as well. Moreover, tables are prepared and displayed for a more physical insight.

**Keywords:** hybrid nanofluid; induced magnetic field; mixed convection; heat generation; peristalsis; cilia beating; Non-Newtonian