Some New Post-Quantum Integral Inequalities Involving Twice (p, q)-Differentiable ψ-Preinvex Functions and Applications

Abstract

The main motivation of this article is derive a new post-quantum integral identity using twice (p, q)-differentiable functions. Using the identity as an auxiliary result, we will obtain some new variants of Hermite–Hadamard’s inequality essentially via the class of ψ-preinvex functions. To support our results, we offer some applications to a special means of positive real numbers and twice (p, q)-differentiable functions that are in absolute value bounded as well.

1. Introduction and Preliminaries

Inequalities play a pivotal role in almost all branches of mathematics. For instance, the inequalities arising from the convexity property of related functions have numerous applications in the study of qualitative theory of differential equations and partial differential equations (see, for example, the papers of [1,2] for more details). In modern analysis, a significant amount of inequalities can be obtained by using the convexity property of the functions. Hermite–Hadamard’s inequality is one of the most studied inequalities pertaining to convexity. This result reads as

$$\mathsf{{\rm Y}}\left(\frac{{\mu }_1+{\mu }_2}{2}\right)\le \frac{1}{{\mu }_2-{\mu }_1}\underset{{\mu }_1}{\overset{{\mu }_2}{\int }}\mathsf{{\rm Y}}\left(x\right)\mathrm{d}x\le \frac{\mathsf{{\rm Y}}\left({\mu }_1\right)+\mathsf{{\rm Y}}\left({\mu }_2\right)}{2}$$

if $\mathsf{{\rm Y}}:[{\mu }_1,{\mu }_2]\mapsto \mathbb{R}$ is a convex function on closed interval $[{\mu }_1,{\mu }_2].$

In recent years, the improvements, generalizations, and variants of Hermite–Hadamard’s inequality have been the subject of much research. In this regard, a variety of novel and innovative approaches have been utilized in obtaining new refinements of Hermite–Hadamard’s inequality. For the first time, Tariboon and Ntouyas [3] obtained a $\mathrm{q}$-analogue of Hermite–Hadamard’s inequality using the concepts of quantum calculus, which is also known as calculus without limits. In quantum calculus, we establish the $\mathrm{q}$-analogues of classical mathematical objects that can be recaptured by taking $\mathrm{q}\to 1^{-}$. Alp et al. [4] obtained a corrected $\mathrm{q}$-analogue of Hermite–Hadamard’s inequality. Noor et al. [5] and Sudsutad et al. [6] derived some more $\mathrm{q}$-analogues of Hermite–Hadamard-like inequalities involving first order $\mathrm{q}$-differentiable convex functions, and Liu and Zhuang [7] established these analogues via second order $\mathrm{q}$-differentiable convex functions. Zhang et al. [8] obtained a new generalized $\mathrm{q}$-integral identity and obtained several new $\mathrm{q}$-analogues of a first order $\mathrm{q}$-differentiable convex function.

Chakarabarti and Jagannathan [9] studied post-quantum calculus, which is another significant generalization of quantum calculus is the post-quantum calculus. In quantum calculus, we deal with a $\mathrm{q}$-number with one base $\mathrm{q}$, but post-quantum calculus includes $\mathrm{p}$ and $\mathrm{q}$-numbers with two independent variables $\mathrm{p}$ and $\mathrm{q}$. Tunç and Gov [10] introduced the concepts of $(\mathrm{p},\mathrm{q})$-derivatives ${}_{{\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}\mathsf{{\rm Y}}\left(x\right)$ and $(\mathrm{p},\mathrm{q})$-integrals on finite intervals $\underset{{\mu }_1}{\overset{x}{\int }}\mathsf{{\rm Y}}\left(\tau \right){}_{{\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau$ for all $x\ne {\mu }_1,$ where $x\in \mathcal{K}\subset \mathbb{R}$, as follows.

Definition 1

([10]). Let $\mathsf{{\rm Y}}:\mathcal{K}\subset \mathbb{R}\mapsto \mathbb{R}$ be a continuous function and let $x\in \mathcal{K}$ and $0<\mathrm{q}<\mathrm{p}\le 1$. The $(\mathrm{p},\mathrm{q})$-derivative on $\mathcal{K}$ of function Υ at x is then defined as

$$\begin{array}{c}\hfill {}_{{\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}\mathsf{{\rm Y}}\left(x\right)=\frac{\mathsf{{\rm Y}}(\mathrm{p}x+(1-\mathrm{p}){\mu }_1)-\mathsf{{\rm Y}}(\mathrm{q}x+(1-\mathrm{q}){\mu }_1)}{(\mathrm{p}-\mathrm{q})(x-{\mu }_1)},x\ne {\mu }_1.\end{array}$$

(1)

Definition 2

([10]). Let $\mathsf{{\rm Y}}:\mathcal{K}\subset \mathbb{R}\mapsto \mathbb{R}$ be a continuous function. The $(\mathrm{p},\mathrm{q})$-integral on $\mathcal{K}$ is then defined as

$$\begin{array}{c}\hfill \underset{{\mu }_1}{\overset{x}{\int }}\mathsf{{\rm Y}}\left(\tau \right){}_{{\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau =(\mathrm{p}-\mathrm{q})(x-{\mu }_1)\sum _{n=0}^{\infty }\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\mathsf{{\rm Y}}\left(\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}x+\left(1-\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\right){\mu }_1\right),\end{array}$$

for $x\in \mathcal{K}$ and $x\ne {\mu }_1.$

Since then, several new variants of classical integral inequalities have been obtained using the concepts of post-quantum calculus. For example, Awan et al. [11] obtained a generalized $(\mathrm{p},\mathrm{q})$-integral identity and obtained several new $(\mathrm{p},\mathrm{q})$-analogues of trapezium-like inequalities. Kunt et al. [12] obtained some $(\mathrm{p},\mathrm{q})$-analogues of Hermite–Hadamard and mid-point type inequalities. Yu et al. [13] derived several new $(\mathrm{p},\mathrm{q})$-analogues of some classical integral inequalities and discussed applications as well.

Definition 3.

A set $\mathcal{K}\subset \mathbb{R}$ is said to be an invex set with respect to the mapping $\zeta :\mathcal{K}\times \mathcal{K}\times [0,1]\mapsto \mathbb{R}$ if $\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi )\in \mathcal{K}$ for every ${\mu }_1,{\mu }_2\in \mathcal{K}$ and $\psi ,\tau \in [0,1].$ The invex set $\mathcal{K}$ is also called an ζ-connected set.

Before we proceed further, let us recall the definition of $\psi$-preinvex functions.

Definition 4

([14]). A function Υ on the invex set $\mathcal{K}$ is said to be ψ-preinvex with respect to $\zeta ({\mu }_2,{\mu }_1,\psi )$ if

$$\begin{array}{c}\hfill \mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))\le \psi (1-\tau )\mathsf{{\rm Y}}\left({\mu }_1\right)+\tau \mathsf{{\rm Y}}\left({\mu }_2\right),\forall {\mu }_1,{\mu }_2\in \mathcal{K},\psi ,\tau \in [0,1].\end{array}$$

Remark 1.

Note the following:

I.

If we take $\zeta ({\mu }_2,{\mu }_1,\psi )={\mu }_2-\psi {\mu }_1$ in Definition 4, then we have the definition of a ψ-convex function, see [15].

II.

If we choose $\psi =1$ in Definition 4, then we obtain the class of classical preinvex functions, see [16].

The main motivation of this article is to derive a new post-quantum integral identity using twice $(\mathrm{p},\mathrm{q})$-differentiable functions. Using the identity as an auxiliary result, we will obtain some new variants of Hermite–Hadamard’s inequality essentially via the class of $\psi$-preinvex functions. To support our results, we also present some applications to a special means of positive real numbers and twice $(\mathrm{p},\mathrm{q})$-differentiable functions that are in absolute value bounded. We hope that the ideas and techniques of this paper will inspire interested readers working in this field.

2. Main Results

In this section, we derive new post-quantum integral identity. This result will be helpful in obtaining main results of this paper.

Lemma 1.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$ (the interior of set $\mathcal{K}$), and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and $(\mathrm{p},\mathrm{q})$-integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Thus,

$$\begin{array}{cc}\hfill & \frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\hfill \\ \hfill & =\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau ){}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi )){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau .\hfill \end{array}$$

Proof. 

Applying Definition 1, we have

$$\begin{array}{cc}\hfill & {}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))\hfill \\ \hfill & ={}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}\left({}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))\right)\hfill \\ \hfill & =\frac{{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\tau \zeta ({\mu }_2,{\mu }_1,\psi ))-{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{q}\tau \zeta ({\mu }_2,{\mu }_1,\psi ))}{\tau (\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi )}\hfill \\ \hfill & =\frac{1}{\tau (\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi )}\left[\frac{\mathsf{{\rm Y}}(\psi {\mu }_1+{\mathrm{p}}^2\tau \zeta ({\mu }_2,{\mu }_1,\psi ))-\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\mathrm{q}\tau \zeta ({\mu }_2,{\mu }_1,\psi ))}{\tau p(\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi )}\right.\hfill \\ \hfill & \left.-\frac{\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\mathrm{q}\tau \zeta ({\mu }_2,{\mu }_1,\psi ))-\mathsf{{\rm Y}}(\psi {\mu }_1+{\mathrm{q}}^2\tau \zeta ({\mu }_2,{\mu }_1,\psi ))}{\tau q(\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi )}\right]\hfill \\ \hfill & =\frac{\mathrm{q}\mathsf{{\rm Y}}(\psi {\mu }_1+{\mathrm{p}}^2\tau \zeta ({\mu }_2,{\mu }_1,\psi ))-(\mathrm{p}+\mathrm{q})\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\mathrm{q}\tau \zeta ({\mu }_2,{\mu }_1,\psi ))+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+{\mathrm{q}}^2\tau \zeta ({\mu }_2,{\mu }_1,\psi ))}{\mathrm{p}\mathrm{q}{\tau }^2{(\mathrm{p}-\mathrm{q})}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}.\hfill \end{array}$$

Now by using Definition 2, we obtain

$$\begin{array}{cc}\hfill & \underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau ){}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi )){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & =\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )\hfill \\ \hfill & \times \frac{\mathrm{q}\mathsf{{\rm Y}}(\psi {\mu }_1+{\mathrm{p}}^2\tau \zeta ({\mu }_2,{\mu }_1,\psi ))-(\mathrm{p}+\mathrm{q})\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{q}\tau \zeta ({\mu }_2,{\mu }_1,\psi ))+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+{\mathrm{q}}^2\tau \zeta ({\mu }_2,{\mu }_1,\psi ))}{{\tau }^2\mathrm{p}\mathrm{q}{(\mathrm{p}-\mathrm{q})}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & =\frac{1}{\mathrm{p}\mathrm{q}(\mathrm{p}-\mathrm{q}){\zeta }^2({\mu }_2,{\mu }_1,\psi )}\left[\mathrm{q}\sum _{n=0}^{\infty }\mathsf{{\rm Y}}\left(\psi {\mu }_1+{\mathrm{p}}^2\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)-(\mathrm{p}+\mathrm{q})\sum _{n=0}^{\infty }\mathsf{{\rm Y}}\left(\psi {\mu }_1+\mathrm{p}\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)\right.\hfill \\ \hfill & \left.+\mathrm{p}\sum _{n=0}^{\infty }\mathsf{{\rm Y}}\left(\psi {\mu }_1+\frac{{\mathrm{q}}^{n+2}}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)\right]\hfill \\ \hfill & -\mathrm{q}\left\{\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi ){\displaystyle \sum _{n=0}^{\infty }}\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\mathsf{{\rm Y}}\left(\psi {\mu }_1+{\mathrm{p}}^2\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)}{\mathrm{p}\mathrm{q}{(\mathrm{p}-\mathrm{q})}^2{\zeta }^3({\mu }_2,{\mu }_1,\psi )}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -\frac{(\mathrm{p}+\mathrm{q})(\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi ){\displaystyle \sum _{n=0}^{\infty }}\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\mathsf{{\rm Y}}\left(\psi {\mu }_1+\mathrm{p}\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)}{\mathrm{p}{\mathrm{q}}^2{(\mathrm{p}-\mathrm{q})}^2{\zeta }^3({\mu }_2,{\mu }_1,\psi )}\hfill \\ \hfill & \left.+\frac{\mathrm{p}(\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi ){\displaystyle \sum _{n=0}^{\infty }}\frac{{\mathrm{q}}^{n+2}}{{\mathrm{p}}^{n+1}}\mathsf{{\rm Y}}\left(\psi {\mu }_1+\frac{{\mathrm{q}}^{n+2}}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)}{\mathrm{p}{\mathrm{q}}^3{(\mathrm{p}-\mathrm{q})}^2{\zeta }^3({\mu }_2,{\mu }_1,\psi )}\right\}\hfill \\ \hfill & =\frac{\mathrm{q}\left[{\displaystyle \sum _{n=0}^{\infty }}\mathsf{{\rm Y}}\left(\psi {\mu }_1+{\mathrm{p}}^2\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)-{\displaystyle \sum _{n=0}^{\infty }}\mathsf{{\rm Y}}\left(\psi {\mu }_1+\mathrm{p}\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)\right]}{\mathrm{p}\mathrm{q}(\mathrm{p}-\mathrm{q}){\zeta }^2({\mu }_2,{\mu }_1,\psi )}\hfill \\ \hfill & -\frac{\mathrm{p}\left[{\displaystyle \sum _{n=0}^{\infty }}\mathsf{{\rm Y}}\left(\psi {\mu }_1+\mathrm{p}\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)-{\displaystyle \sum _{n=0}^{\infty }}\mathsf{{\rm Y}}\left(\psi {\mu }_1+\frac{{\mathrm{q}}^{n+2}}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)\right]}{\mathrm{p}\mathrm{q}(\mathrm{p}-\mathrm{q}){\zeta }^2({\mu }_2,{\mu }_1,\psi )}\hfill \\ \hfill & -\mathrm{q}\left\{\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi ){\displaystyle \sum _{n=0}^{\infty }}\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\mathsf{{\rm Y}}\left(\psi {\mu }_1+{\mathrm{p}}^2\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)}{\mathrm{p}\mathrm{q}{(\mathrm{p}-\mathrm{q})}^2{\zeta }^3({\mu }_2,{\mu }_1,\psi )}\right.\hfill \\ \hfill & -\frac{{\mathrm{p}}^2(\mathrm{p}+\mathrm{q})(\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi ){\displaystyle \sum _{n=0}^{\infty }}\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+2}}\mathsf{{\rm Y}}\left(\psi {\mu }_1+{\mathrm{p}}^2\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+2}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)}{\mathrm{p}{\mathrm{q}}^2{(\mathrm{p}-\mathrm{q})}^2{\zeta }^3({\mu }_2,{\mu }_1,\psi )}\hfill \\ \hfill & \left.+\frac{{\mathrm{p}}^3(\mathrm{p}-\mathrm{q})\zeta ({\mu }_2,{\mu }_1,\psi ){\displaystyle \sum _{n=0}^{\infty }}\frac{{\mathrm{q}}^{n+2}}{{\mathrm{p}}^{n+3}}\mathsf{{\rm Y}}\left(\psi {\mu }_1+{\mathrm{p}}^2\frac{{\mathrm{q}}^{n+2}}{{\mathrm{p}}^{n+3}}\zeta ({\mu }_2,{\mu }_1,\psi )\right)}{\mathrm{p}{\mathrm{q}}^3{(\mathrm{p}-\mathrm{q})}^2{\zeta }^3({\mu }_2,{\mu }_1,\psi )}\right\}\hfill \\ \hfill & =\frac{\mathrm{q}\left[\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )-\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)\right]-\mathrm{p}\left[\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{q}\zeta ({\mu }_2,{\mu }_1,\psi ))-\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)\right]}{\mathrm{p}\mathrm{q}(\mathrm{p}-\mathrm{q}){\zeta }^2({\mu }_2,{\mu }_1,\psi )}\hfill \\ \hfill & -\frac{\mathrm{p}+\mathrm{q}}{{\mathrm{p}}^3{\mathrm{q}}^2{\zeta }^3({\mu }_2,{\mu }_1,\psi )}{\displaystyle \underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}}\mathsf{{\rm Y}}\left(x\right){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau -\frac{{\mathrm{q}}^2+\mathrm{p}\mathrm{q}-\mathrm{p}}{\mathrm{p}{\mathrm{q}}^2(\mathrm{p}-\mathrm{q}){\zeta }^2({\mu }_2,{\mu }_1,\psi )}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi ))\hfill \\ \hfill & +\frac{\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{q}\zeta ({\mu }_2,{\mu }_1,\psi ))}{\mathrm{q}(\mathrm{p}-\mathrm{q}){\zeta }^2({\mu }_2,{\mu }_1,\psi )}\hfill \\ \hfill & =\frac{\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)}{\mathrm{p}\mathrm{q}{\zeta }^2({\mu }_2,{\mu }_1,\psi )}+\frac{\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi ))}{{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}-\frac{\mathrm{p}+\mathrm{q}}{{\mathrm{p}}^3{\mathrm{q}}^2{\zeta }^3({\mu }_2,{\mu }_1,\psi )}{\displaystyle \underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x.\hfill \end{array}$$

Multiplying both sides of the above equality by $\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}$, we obtain the required result. □

Using Lemma 1, we can obtain the following new results.

Theorem 1.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and $(\mathrm{p},\mathrm{q})$-integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Assume that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}|$ is a ψ-preinvex function. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )(\psi ({\mathrm{p}}^4-{\mathrm{p}}^3+{\mathrm{p}}^2{\mathrm{q}}^2)|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right)|+{\mathrm{p}}^3|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|)}{{(\mathrm{p}+\mathrm{q})}^2({\mathrm{p}}^2+{\mathrm{q}}^2)({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}.\hfill \end{array}$$

(2)

Proof. 

Using Lemma 1, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}|$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\left(\psi |{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right)|\underset{0}{\overset{1}{\int }}\tau (1-\tau )(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau +|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|\underset{0}{\overset{1}{\int }}{\tau }^2(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)\hfill \\ \hfill & =\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )(\psi ({\mathrm{p}}^4-{\mathrm{p}}^3+{\mathrm{p}}^2{\mathrm{q}}^2)|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right)|+{\mathrm{p}}^3|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|)}{{(\mathrm{p}+\mathrm{q})}^2({\mathrm{p}}^2+{\mathrm{q}}^2)({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}.\hfill \end{array}$$

This completes the proof. □

Theorem 2.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Suppose that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$ is a ψ-preinvex function for $r\ge 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}{\left(\psi d_1|{}_{{\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left(\psi {\mu }_1\right){|}^r+d_2{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

(3)

where

$$d_1:=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }\left(\frac{{\mathrm{q}}^{2n}}{{\mathrm{p}}^{2n+2}}-\frac{{\mathrm{q}}^{3n}}{{\mathrm{p}}^{3n+3}}\right){\left(1-\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\right)}^r$$

and

$$d_2:=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }\frac{{\mathrm{q}}^{3n}}{{\mathrm{p}}^{3n+3}}{\left(1-\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\right)}^r.$$

Proof. 

Using Lemma 1, the power mean inequality, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}\tau {}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{1-\frac{1}{r}}{\left(\underset{0}{\overset{1}{\int }}\tau {(1-\mathrm{q}\tau )}^r{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\frac{1}{\mathrm{p}+\mathrm{q}}\right)}^{1-\frac{1}{r}}\left(\psi |{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r\underset{0}{\overset{1}{\int }}\tau (1-\tau ){(1-\mathrm{q}\tau )}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right.\hfill \\ \hfill & {\left.+|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right){|}^r\underset{0}{\overset{1}{\int }}{\tau }^2{(1-\mathrm{q}\tau )}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}{\left(\psi d_1|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+d_2{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 3.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Assume that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$ is a ψ-preinvex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{h}^{\frac{1}{s}}{\left(\frac{\psi ({\mathrm{q}}^2+{\mathrm{p}}^2+\mathrm{p}\mathrm{q}-\mathrm{p}-\mathrm{q})|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+(\mathrm{p}+\mathrm{q}){|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r}{(\mathrm{p}+\mathrm{q})({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

(4)

where

$$h:=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }\frac{{\mathrm{q}}^{2n}}{{\mathrm{p}}^{2n+2}}{\left(1-\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\right)}^s.$$

Proof. 

Using Lemma 1, Hölder’s inequality, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}\tau {(1-\mathrm{q}\tau )}^s{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}\tau {|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}\tau {(1-\mathrm{q}\tau )}^s{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{s}}\hfill \\ \hfill & \times {\left(\psi |{}_{{\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left(\psi {\mu }_1\right){|}^r\underset{0}{\overset{1}{\int }}\tau (1-\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau +{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\underset{0}{\overset{1}{\int }}{\tau }^2{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{h}^{\frac{1}{s}}{\left(\frac{\psi ({\mathrm{q}}^2+{\mathrm{p}}^2+\mathrm{p}\mathrm{q}-\mathrm{p}-\mathrm{q})|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+(\mathrm{p}+\mathrm{q}){|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r}{(\mathrm{p}+\mathrm{q})({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 4.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Suppose that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$ is a ψ-preinvex function for $r\ge 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\left(\psi k_1|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+k_2{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

(5)

where

$$k_1:=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }{\left(\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\right)}^{r+1}\left(1-\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\right){\left(1-\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\right)}^r$$

and

$$k_2:=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }{\left(\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\right)}^{r+3}{\left(1-\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\right)}^r.$$

Proof. 

Using Lemma 1, the power mean inequality, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}1{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{1-\frac{1}{r}}{\left(\underset{0}{\overset{1}{\int }}{\tau }^r{(1-\mathrm{q}\tau )}^r{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\left(\psi |{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r\underset{0}{\overset{1}{\int }}{\tau }^r(1-\tau ){(1-\mathrm{q}\tau )}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right.\hfill \\ \hfill & {\left.+|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right){|}^r\underset{0}{\overset{1}{\int }}{\tau }^{r+2}{(1-\mathrm{q}\tau )}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\left(\psi k_1|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+k_2{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 5.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Assume that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$ is a ψ-preinvex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\omega }^{\frac{1}{s}}{\left(\frac{\psi (\mathrm{p}+\mathrm{q}-1)|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r}{(\mathrm{p}+\mathrm{q})}\right)}^{\frac{1}{r}},\hfill \end{array}$$

(6)

where

$$\omega :=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }{\left(\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\right)}^{s+1}{\left(1-\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\right)}^s.$$

Proof. 

Using Lemma 1, Hölder’s inequality, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}{\tau }^s{(1-\mathrm{q}\tau )}^s{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\hfill \\ \hfill & \times {\left(\underset{0}{\overset{1}{\int }}{\tau }^s{(1-\mathrm{q}\tau )}^s{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{s}}{\left(\psi |{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r\underset{0}{\overset{1}{\int }}(1-\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau +{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\underset{0}{\overset{1}{\int }}\tau {}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\omega }^{\frac{1}{s}}{\left(\frac{\psi (\mathrm{p}+\mathrm{q}-1)|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r}{(\mathrm{p}+\mathrm{q})}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 6.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Suppose that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$ is a ψ-preinvex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\left(\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{s+1}-{\mathrm{q}}^{s+1}}\right)}^{\frac{1}{s}}{\left(\psi {\Delta }_1|{}_{{\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+{\Delta }_2{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

(7)

where

$${\Delta }_1:=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }\left(\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}-\frac{{\mathrm{q}}^{2n}}{{\mathrm{p}}^{2n+2}}\right){\left(1-\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\right)}^r$$

and

$${\Delta }_2:=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }\frac{{\mathrm{q}}^{2n}}{{\mathrm{p}}^{2n+2}}{\left(1-\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\right)}^r.$$

Proof. 

Using Lemma 1, Hölder’s inequality, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}{\tau }^s{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}{(1-\mathrm{q}\tau )}^r{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{s+1}-{\mathrm{q}}^{s+1}}\right)}^{\frac{1}{s}}\left(\psi |{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r\underset{0}{\overset{1}{\int }}(1-\tau ){(1-\mathrm{q}\tau )}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right.\hfill \\ \hfill & {\left.+|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right){|}^r\underset{0}{\overset{1}{\int }}\tau {(1-\mathrm{q}\tau )}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\left(\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{s+1}-{\mathrm{q}}^{s+1}}\right)}^{\frac{1}{s}}{\left(\psi {\Delta }_1|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+{\Delta }_2{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 7.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Assume that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$ is a ψ-preinvex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\gamma }^{\frac{1}{s}}{\left(\psi \left(\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+1}-{\mathrm{q}}^{r+1}}-\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}\right)|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

(8)

where

$$\gamma :=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}{\left(1-\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\right)}^s.$$

Proof. 

Using Lemma 1, Hölder’s inequality, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}(1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}{(1-\mathrm{q}\tau )}^s{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}{\tau }^r{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}{(1-\mathrm{q}\tau )}^s{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{s}}\left(\psi |{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r\underset{0}{\overset{1}{\int }}{\tau }^r(1-\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right.\hfill \\ \hfill & {\left.+|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right){|}^r\underset{0}{\overset{1}{\int }}{\tau }^{r+1}{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\gamma }^{\frac{1}{s}}{\left(\psi \left(\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+1}-{\mathrm{q}}^{r+1}}-\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}\right)|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 8.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Suppose that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$ is a ψ-preinvex function for $r\ge 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{{\mathrm{p}}^{2-\frac{1}{r}}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}{\left(\frac{{\mathrm{p}}^2}{{\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2}\right)}^{1-\frac{1}{r}}{\left(\frac{\psi ({\mathrm{p}}^4-{\mathrm{p}}^3+{\mathrm{p}}^2{\mathrm{q}}^2)|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+{\mathrm{p}}^3{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r}{(\mathrm{p}+\mathrm{q})({\mathrm{p}}^2+{\mathrm{q}}^2)({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(9)

Proof. 

Using Lemma 1, the power mean inequality, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{1-\frac{1}{r}}{\left(\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau ){|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{1-\frac{1}{r}}\hfill \\ \hfill & \times {\left(\psi |{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r\underset{0}{\overset{1}{\int }}\tau (1-\tau )(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau +{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\underset{0}{\overset{1}{\int }}{\tau }^2(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{\mathrm{p}}^{2-\frac{1}{r}}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}{\left(\frac{{\mathrm{p}}^2}{{\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2}\right)}^{1-\frac{1}{r}}{\left(\frac{\psi ({\mathrm{p}}^4-{\mathrm{p}}^3+{\mathrm{p}}^2{\mathrm{q}}^2)|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+{\mathrm{p}}^3{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r}{(\mathrm{p}+\mathrm{q})({\mathrm{p}}^2+{\mathrm{q}}^2)({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 9.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Assume that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$ is a ψ-preinvex function for $r\ge 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{{\mathrm{p}}^{2-\frac{1}{r}}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}\left(\psi \left[\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+1}-{\mathrm{q}}^{r+1}}-\frac{(\mathrm{p}-\mathrm{q})(1+\mathrm{q})}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}+\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^{r+3}-{\mathrm{q}}^{r+3}}\right]{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right)|}^r\right.\hfill \\ \hfill & {\left.+\left[\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}-\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^{r+3}-{\mathrm{q}}^{r+3}}\right]{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(10)

Proof. 

Using Lemma 1, the power mean inequality, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{1-\frac{1}{r}}{\left(\underset{0}{\overset{1}{\int }}{\tau }^r(1-\mathrm{q}\tau ){|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{1-\frac{1}{r}}\hfill \\ \hfill & \times {\left(\psi |{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r\underset{0}{\overset{1}{\int }}{\tau }^r(1-\tau )(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau +{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\underset{0}{\overset{1}{\int }}{\tau }^{r+1}(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{\mathrm{p}}^{2-\frac{1}{r}}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}\left(\psi \left[\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+1}-{\mathrm{q}}^{r+1}}-\frac{(\mathrm{p}-\mathrm{q})(1+\mathrm{q})}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}+\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^{r+3}-{\mathrm{q}}^{r+3}}\right]{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right)|}^r\right.\hfill \\ \hfill & {\left.+\left[\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}-\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^{r+3}-{\mathrm{q}}^{r+3}}\right]{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 10.

Let $\mathsf{{\rm Y}}:\mathcal{K}\mapsto \mathbb{R}$ be a twice $(\mathrm{p},\mathrm{q})$-differentiable function on ${\mathcal{K}}^{\circ }$, and let ${}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}$ be continuous and integrable on $\mathcal{K},$ where $0<\mathrm{q}<\mathrm{p}\le 1.$ Suppose that $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$ is a ψ-preinvex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\lambda }^{\frac{1}{s}}{\left(\frac{\psi ({\mathrm{p}}^3-{\mathrm{p}}^2+\mathrm{p}{\mathrm{q}}^2+{\mathrm{p}}^2\mathrm{q})|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+{\mathrm{p}}^2{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r}{(\mathrm{p}+\mathrm{q})({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

(11)

where

$$\lambda :=(\mathrm{p}-\mathrm{q})\sum _{n=0}^{\infty }{\left(\frac{{\mathrm{q}}^n}{{\mathrm{p}}^{n+1}}\right)}^{s+1}{\left(1-\frac{{\mathrm{q}}^{n+1}}{{\mathrm{p}}^{n+1}}\right)}^s.$$

Proof. 

Using Lemma 1, Hölder’s inequality, the $\psi$-preinvexity of $|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2{\mathsf{{\rm Y}}|}^r$, and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau )|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}{\tau }^s(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}(1-\mathrm{q}\tau ){|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}(\psi {\mu }_1+\tau \zeta ({\mu }_2,{\mu }_1,\psi ))|}^r{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}{\left(\underset{0}{\overset{1}{\int }}{\tau }^s{(1-\mathrm{q}\tau )}^s{}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{s}}\hfill \\ \hfill & \times {\left(\psi |{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r\underset{0}{\overset{1}{\int }}(1-\tau )(1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau +{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r\underset{0}{\overset{1}{\int }}\tau (1-\mathrm{q}\tau ){}_0{\mathrm{d}}_{\mathrm{p},\mathrm{q}}\tau \right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\lambda }^{\frac{1}{s}}{\left(\frac{\psi ({\mathrm{p}}^3-{\mathrm{p}}^2+\mathrm{p}{\mathrm{q}}^2+{\mathrm{p}}^2\mathrm{q})|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_1\right){|}^r+{\mathrm{p}}^2{|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}\left({\mu }_2\right)|}^r}{(\mathrm{p}+\mathrm{q})({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

3. Applications

In this section, we will discuss some applications regarding our main results given in Section 2 for special means and bounded functions as well.

3.1. Application to Special Means

First, let us denote

$${\mathcal{M}}^{\diamond }:=\frac{\mathrm{q}{((1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)}^n-(\mathrm{p}+\mathrm{q}){((1-\mathrm{p}\mathrm{q}){\mu }_1+\mathrm{p}\mathrm{q}{\mu }_2)}^n+\mathrm{p}{((1-{\mathrm{q}}^2){\mu }_1+{\mathrm{q}}^2{\mu }_2)}^n}{\mathrm{p}\mathrm{q}{(\mathrm{p}-\mathrm{q})}^2{({\mu }_2-{\mu }_1)}^2},$$

where $0<{\mu }_1<{\mu }_2$ are real numbers and $0<\mathrm{q}<\mathrm{p}\le 1.$

• The arithmetic mean is defined as

  $$\mathcal{A}({\mu }_1,{\mu }_2):=\frac{{\mu }_1+{\mu }_2}{2}.$$
• The generalized logarithmic mean is given by

  $$\begin{array}{c}\hfill {\mathcal{L}}_n({\mu }_1,{\mu }_2):={\left[\frac{{\mu }_2^{n+1}-{\mu }_1^{n+1}}{(n+1)({\mu }_2-{\mu }_1)}\right]}^{\frac{1}{n}},n\in \mathbb{R}\setminus \{-1,0\}.\end{array}$$

Using the above special means, we can establish the following inequalities.

Proposition 1.

Let $0<{\mu }_1<{\mu }_2$, $n>1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{({\mu }_2-{\mu }_1)}^2(({\mathrm{p}}^4-{\mathrm{p}}^3+{\mathrm{p}}^2{\mathrm{q}}^2)n(n-1){\mu }_1^{n-2}+{\mathrm{p}}^3|M^{\diamond }|)}{{(\mathrm{p}+\mathrm{q})}^2({\mathrm{p}}^2+{\mathrm{q}}^2)({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}.\hfill \end{array}$$

(12)

Proof. 

The proof directly follows from Theorem 1, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

Proposition 2.

Let $0<{\mu }_1<{\mu }_2$, $n>1,r\ge 1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2({\mu }_2-{\mu }_1)}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}{\left(d_{1}n(n-1){\mu }_1^{r(n-2)}+d_2{|{\mathcal{M}}^{\diamond }|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

(13)

Proof. 

The proof directly follows from Theorem 2, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

Proposition 3.

Let $0<{\mu }_1<{\mu }_2$, $n>1,r>1,\frac{1}{s}+\frac{1}{r}=1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2({\mu }_2-{\mu }_1)2}{(\mathrm{p}+\mathrm{q})}{h}^{\frac{1}{s}}{\left(\frac{({\mathrm{q}}^2+{\mathrm{p}}^2+\mathrm{p}\mathrm{q}-\mathrm{p}-\mathrm{q})n(n-1){\mu }_1^{r(n-2)}+(\mathrm{p}+\mathrm{q}){|{\mathcal{M}}^{\diamond }|}^r}{(\mathrm{p}+\mathrm{q})({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(14)

Proof. 

The proof directly follows from Theorem 3, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

Proposition 4.

Let $0<{\mu }_1<{\mu }_2$, $n>1,r\ge 1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{({\mu }_2-{\mu }_1)}^2}{(\mathrm{p}+\mathrm{q})}{\left(n(n-1)k_1{\mu }_1^{r(n-2)}+k_2{|{\mathcal{M}}^{\diamond }|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(15)

Proof. 

The proof directly follows from Theorem 4, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

Proposition 5.

Let $0<{\mu }_1<{\mu }_2$, $n>1,r>1,\frac{1}{s}+\frac{1}{r}=1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{({\mu }_2-{\mu }_1)}^2}{(\mathrm{p}+\mathrm{q})}{\omega }^{\frac{1}{s}}{\left(\frac{n(n-1)(\mathrm{p}+\mathrm{q}-1){\mu }_1^{r(n-2)}+{|{\mathcal{M}}^{\diamond }|}^r}{(\mathrm{p}+\mathrm{q})}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(16)

Proof. 

The proof directly follows from Theorem 5, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

Proposition 6.

Let $0<{\mu }_1<{\mu }_2$, $n>1,r>1,\frac{1}{s}+\frac{1}{r}=1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{({\mu }_2-{\mu }_1)}^2}{(\mathrm{p}+\mathrm{q})}{\left(\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{s+1}-{\mathrm{q}}^{s+1}}\right)}^{\frac{1}{s}}{\left(n(n-1){\Delta }_1{\mu }_1^{r(n-2)}+{\Delta }_2{|{\mathcal{M}}^{\diamond }|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(17)

Proof. 

The proof directly follows from Theorem 6, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

Proposition 7.

Let $0<{\mu }_1<{\mu }_2$, $n>1,r>1,\frac{1}{s}+\frac{1}{r}=1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{({\mu }_2-{\mu }_1)}^2}{(\mathrm{p}+\mathrm{q})}{\gamma }^{\frac{1}{s}}{\left(\left(\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+1}-{\mathrm{q}}^{r+1}}-\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}\right){\mu }_1^{r(n-2)}+\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}{|{\mathcal{M}}^{\diamond }|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(18)

Proof. 

The proof directly follows from Theorem 7, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

Proposition 8.

Let $0<{\mu }_1<{\mu }_2$, $n>1,r\ge 1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{{\mathrm{p}}^{2-\frac{1}{r}}{\mathrm{q}}^2{({\mu }_2-{\mu }_1)}^2}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}{\left(\frac{{\mathrm{p}}^2}{{\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2}\right)}^{1-\frac{1}{r}}{\left(\frac{n(n-1)({\mathrm{p}}^4-{\mathrm{p}}^3+{\mathrm{p}}^2{\mathrm{q}}^2){\mu }_1^{r(n-2)}+{\mathrm{p}}^3{|{\mathcal{M}}^{\diamond }|}^r}{(\mathrm{p}+\mathrm{q})({\mathrm{p}}^2+{\mathrm{q}}^2)({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(19)

Proof. 

The proof directly follows from Theorem 8, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

Proposition 9.

Let $0<{\mu }_1<{\mu }_2$, $n>1,r\ge 1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{{\mathrm{p}}^{2-\frac{1}{r}}{\mathrm{q}}^2{({\mu }_2-{\mu }_1)}^2}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}\left(\left[\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+1}-{\mathrm{q}}^{r+1}}-\frac{(\mathrm{p}-\mathrm{q})(1+\mathrm{q})}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}+\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^{r+3}-{\mathrm{q}}^{r+3}}\right]n(n-1){\mu }_1^{r(n-2)}\right.\hfill \\ \hfill & {\left.+\left[\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}-\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^{r+3}-{\mathrm{q}}^{r+3}}\right]{|{\mathcal{M}}^{\diamond }|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

(20)

Proof. 

The proof directly follows from Theorem 9, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

Proposition 10.

Let $0<{\mu }_1<{\mu }_2$, $n>1,r>1,\frac{1}{s}+\frac{1}{r}=1$ and $0<\mathrm{q}<\mathrm{p}\le 1$. Thus,

$$\begin{array}{cc}\hfill & \left|\frac{2}{\mathrm{p}+\mathrm{q}}\mathcal{A}\left({\mu }_1^n,\mathrm{p}{((1-\mathrm{p}){\mu }_1+\mathrm{p}{\mu }_2)}^n\right)-\frac{(n+1)(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^2({\mathrm{p}}^{n+1}-{\mathrm{q}}^{n+1})}{\mathcal{L}}_n^n({\mu }_1,(1-{\mathrm{p}}^2){\mu }_1+{\mathrm{p}}^2{\mu }_2)\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{({\mu }_2-{\mu }_1)}^2}{(\mathrm{p}+\mathrm{q})}{\lambda }^{\frac{1}{s}}{\left(\frac{n(n-1)({\mathrm{p}}^3-{\mathrm{p}}^2+\mathrm{p}{\mathrm{q}}^2+{\mathrm{p}}^2\mathrm{q}){\mu }_1^{r(n-2)}+{\mathrm{p}}^2{|{\mathcal{M}}^{\diamond }|}^r}{(\mathrm{p}+\mathrm{q})({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

(21)

Proof. 

The proof directly follows from Theorem 10, applying $\mathsf{{\rm Y}}\left(x\right)=x^n,\psi =1$ and $\zeta ({\mu }_2,{\mu }_1)={\mu }_2-{\mu }_1$. □

3.2. Application to Bounded Functions

We suppose that the following condition is satisfied:

$$|{}_{\psi {\mu }_1}{\mathcal{D}}_{\mathrm{p},\mathrm{q}}^2\mathsf{{\rm Y}}|\le ϝ,$$

(22)

which means that the twice $(\mathrm{p},\mathrm{q})$-differentiable function $\mathsf{{\rm Y}}$ is in absolute value bounded from the positive real number $ϝ.$ Applying the above condition, we are in a position to derive some new interesting inequalities using our main results.

Proposition 11.

Under the conditions of Theorem 1, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )(\psi ({\mathrm{p}}^4-{\mathrm{p}}^3+{\mathrm{p}}^2{\mathrm{q}}^2)+{\mathrm{p}}^3)ϝ}{{(\mathrm{p}+\mathrm{q})}^2({\mathrm{p}}^2+{\mathrm{q}}^2)({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}.\hfill \end{array}$$

(23)

Proposition 12.

Under the conditions of Theorem 2, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}{\left(\psi d_1+d_2\right)}^{\frac{1}{r}}ϝ.\hfill \end{array}$$

(24)

Proposition 13.

Under the conditions of Theorem 3, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{h}^{\frac{1}{s}}{\left(\frac{\psi ({\mathrm{q}}^2+{\mathrm{p}}^2+\mathrm{p}\mathrm{q}-\mathrm{p}-\mathrm{q})+\mathrm{p}+\mathrm{q}}{(\mathrm{p}+\mathrm{q})({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}}ϝ.\hfill \end{array}$$

(25)

Proposition 14.

Under the conditions of Theorem 4, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\left(\psi k_1+k_2\right)}^{\frac{1}{r}}ϝ.\hfill \end{array}$$

(26)

Proposition 15.

Under the conditions of Theorem 5, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\omega }^{\frac{1}{s}}{\left(\frac{\psi (\mathrm{p}+\mathrm{q}-1)+1}{(\mathrm{p}+\mathrm{q})}\right)}^{\frac{1}{r}}ϝ.\hfill \end{array}$$

(27)

Proposition 16.

Under the conditions of Theorem 6, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\left(\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{s+1}-{\mathrm{q}}^{s+1}}\right)}^{\frac{1}{s}}{\left(\psi {\Delta }_1+{\Delta }_2\right)}^{\frac{1}{r}}ϝ.\hfill \end{array}$$

(28)

Proposition 17.

Under the conditions of Theorem 7, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\gamma }^{\frac{1}{s}}{\left(\psi \left(\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+1}-{\mathrm{q}}^{r+1}}-\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}\right)+\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}\right)}^{\frac{1}{r}}ϝ.\hfill \end{array}$$

(29)

Proposition 18.

Under the conditions of Theorem 8, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{{\mathrm{p}}^{2-\frac{1}{r}}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}{\left(\frac{{\mathrm{p}}^2}{{\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2}\right)}^{1-\frac{1}{r}}{\left(\frac{\psi ({\mathrm{p}}^4-{\mathrm{p}}^3+{\mathrm{p}}^2{\mathrm{q}}^2)+{\mathrm{p}}^3}{(\mathrm{p}+\mathrm{q})({\mathrm{p}}^2+{\mathrm{q}}^2)({\mathrm{q}}^2+\mathrm{p}\mathrm{q}+{\mathrm{p}}^2)}\right)}^{\frac{1}{r}}ϝ.\hfill \end{array}$$

(30)

Proposition 19.

Under the conditions of Theorem 9, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{{\mathrm{p}}^{2-\frac{1}{r}}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{{(\mathrm{p}+\mathrm{q})}^{2-\frac{1}{r}}}\left(\psi \left[\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+1}-{\mathrm{q}}^{r+1}}-\frac{(\mathrm{p}-\mathrm{q})(1+\mathrm{q})}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}+\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^{r+3}-{\mathrm{q}}^{r+3}}\right]\right.\hfill \\ \hfill & {\left.+\left[\frac{\mathrm{p}-\mathrm{q}}{{\mathrm{p}}^{r+2}-{\mathrm{q}}^{r+2}}-\frac{\mathrm{q}(\mathrm{p}-\mathrm{q})}{{\mathrm{p}}^{r+3}-{\mathrm{q}}^{r+3}}\right]\right)}^{\frac{1}{r}}ϝ.\hfill \end{array}$$

(31)

Proposition 20.

Under the conditions of Theorem 10, the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\mathrm{q}\mathsf{{\rm Y}}\left(\psi {\mu }_1\right)+\mathrm{p}\mathsf{{\rm Y}}(\psi {\mu }_1+\mathrm{p}\zeta ({\mu }_2,{\mu }_1,\psi )}{\mathrm{p}+\mathrm{q}}-\frac{1}{{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}\underset{\psi {\mu }_1}{\overset{\psi {\mu }_1+{\mathrm{p}}^2\zeta ({\mu }_2,{\mu }_1,\psi )}{\int }}\mathsf{{\rm Y}}\left(x\right){}_{\psi {\mu }_1}{\mathrm{d}}_{\mathrm{p},\mathrm{q}}x\right|\hfill \\ \hfill & \le \frac{\mathrm{p}{\mathrm{q}}^2{\zeta }^2({\mu }_2,{\mu }_1,\psi )}{(\mathrm{p}+\mathrm{q})}{\lambda }^{\frac{1}{s}}{\left(\frac{\psi ({\mathrm{p}}^3-{\mathrm{p}}^2+\mathrm{p}{\mathrm{q}}^{}}{}\right)}^{}\hfill \end{array}$$