Some New Parameterized Quantum Fractional Integral Inequalities Involving s-Convex Functions and Applications

Abstract

Convexity performs the appropriate role in the theoretical study of inequalities according to the nature and behavior. Its significance is raised by the strong connection between symmetry and convexity. In this article, we consider a new parameterized quantum fractional integral identity. By applying this identity, we obtain as main results some integral inequalities of trapezium, midpoint and Simpson’s type pertaining to s-convex functions. Moreover, we deduce several special cases, which are discussed in detail. To validate our theoretical findings, an example and application to special means of positive real numbers are presented. Numerical analysis investigation shows that the mixed fractional calculus with quantum calculus give better estimates compared with fractional calculus or quantum calculus separately.

1. Introduction

Integral inequalities are very useful tools for finding estimations and they can be applied in different fields of mathematics, such as FC and DFC etc., see [1,2,3,4,5,6].

Convexity study is very crucial for the theoretical behavior of mathematical inequalities, e.g., [7,8,9]. For some other theoretical studies of inequalities on different types of convex functions, see e.g., s-geometrically [10], $\mathrm{GA}$-convex [11], $\mathrm{MT}$-convex [12], $(\alpha ,m)$-convex [13], $\mathrm{F}$-convex [14], ${\eta }_{\psi }$-convex [15]; a new class of convexity [16] and many other types can be found in [17]. There exist strong linkages and expansive features between the different disciplines of convexity and symmetric function, including probability theory, convex functions, and the geometry of convex functions on convex sets.

Definition 1

([17]). A function $\mathrm{Y}:\mathrm{I}\subseteq \mathbb{R}\to \mathbb{R}$ is said to be convex, if

$$\mathrm{Y}(\varsigma {\vartheta }_1+(1-\varsigma ){\vartheta }_2)\le \varsigma \mathrm{Y}\left({\vartheta }_1\right)+(1-\varsigma )\mathrm{Y}\left({\vartheta }_2\right),$$

holds for all ${\vartheta }_1,{\vartheta }_2\in \mathrm{I}$, and $\varsigma \in [0,1]$. Likewise Υ is concave if $-\mathrm{Y}$ is convex.

Definition 2

([18]). A function $\mathrm{Y}:\mathrm{I}\subseteq \mathbb{R}\to \mathbb{R}$ is said to be s-convex (in the second sense) for some fixed $s\in (0,1],$ if

$$\mathrm{Y}(\varsigma {\vartheta }_1+(1-\varsigma ){\vartheta }_2)\le {\varsigma }^s\mathrm{Y}\left({\vartheta }_1\right)+{(1-\varsigma )}^s\mathrm{Y}\left({\vartheta }_2\right),$$

holds for all ${\vartheta }_1,{\vartheta }_2\in \mathrm{I}$ and $\varsigma \in [0,1]$.

Furthermore, for convex functions and its types many basic inequalities are found such as H-H type [19], Ostrowski type [20], Simpson type [21,22,23,24,25,26], Hardy type [27], Olsen type [28], Gagliardo–Nirenberg type [29], Opial type [30] and Rozanova type [31]. The symmetrical behavior manifests itself naturally in the majority of the aforementioned inequalities.

In this paper, we are focused on three basic integral inequalities, respectively, H-H (trapezium-type inequality), Ostrowski (midpoint-type inequality as a special case) and Simpson types as follows:

Theorem 1

((H-H type inequality) [32]). Suppose that $\mathrm{Y}:\left[{\vartheta }_1,{\vartheta }_2\right]\to \mathbb{R}$ is a convex function, then the following double inequality holds true:

$$\begin{array}{cc}\hfill \mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)& \le \frac{1}{{\vartheta }_2-{\vartheta }_1}{\int }_{{\vartheta }_1}^{{\vartheta }_2}\mathrm{Y}\left(\varsigma \right)d\varsigma \le \frac{\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left({\vartheta }_2\right)}{2}.\hfill \end{array}$$

(1)

Theorem 2

((Ostrowski-type inequality) [33]). Assume that $\mathrm{Y}:\mathrm{I}\subseteq \mathbb{R}\to \mathbb{R}$ is a differentiable function on $\mathrm{I}$ and let ${\vartheta }_1,{\vartheta }_2\in {\mathrm{I}}^{\circ }$ (the interior of $\mathrm{I}$) with ${\vartheta }_1<{\vartheta }_2.$ If $\left|{\mathrm{Y}}^{\prime }\left(\xi \right)\right|\le \mathcal{M}$ for all $\xi \in [{\vartheta }_1,{\vartheta }_2],$ then the following inequality holds true:

$$\begin{array}{c}\hfill \left|\mathrm{Y}\left(\xi \right)-\frac{1}{{\vartheta }_2-{\vartheta }_1}{\int }_{{\vartheta }_1}^{{\vartheta }_2}\mathrm{Y}\left(\varsigma \right)d\varsigma \right|\le \mathcal{M}({\vartheta }_2-{\vartheta }_1)\left[\frac{1}{4}+\frac{{\left(\xi -\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^2}{{({\vartheta }_2-{\vartheta }_1)}^2}\right].\end{array}$$

(2)

Choosing $\xi =\frac{{\vartheta }_1+{\vartheta }_2}{2}$ in (2), we obtain the following midpoint-type inequality:

$$\begin{array}{c}\hfill \left|\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)-\frac{1}{{\vartheta }_2-{\vartheta }_1}{\int }_{{\vartheta }_1}^{{\vartheta }_2}\mathrm{Y}\left(\varsigma \right)d\varsigma \right|\le \frac{\mathcal{M}({\vartheta }_2-{\vartheta }_1)}{4}.\end{array}$$

(3)

Theorem 3

((Simpson-type inequality) [34]). Suppose that $\mathrm{Y}:[{\vartheta }_1,{\vartheta }_2]\to \mathbb{R}$ is a four times continuous and differentiable function on $({\vartheta }_1,{\vartheta }_2)$ such that $\parallel {\mathrm{Y}}^{\left(4\right)}{\parallel }_{\infty }:={sup}_{\varsigma \in ({\vartheta }_1,{\vartheta }_2)}\left|{\mathrm{Y}}^{\left(4\right)}\left(\varsigma \right)\right|<\infty$ with ${\vartheta }_1<{\vartheta }_2$. Then, the following inequality holds true:

$$\begin{array}{c}\hfill \left|\frac{1}{6}\left[\mathrm{Y}\left({\vartheta }_1\right)+4\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\mathrm{Y}\left({\vartheta }_2\right)\right]-\frac{1}{{\vartheta }_2-{\vartheta }_1}{\int }_{{\vartheta }_1}^{{\vartheta }_2}\mathrm{Y}\left(\varsigma \right)d\varsigma \right|\le \frac{1}{2880}{({\vartheta }_2-{\vartheta }_1)}^4{\parallel {\mathrm{Y}}^{\left(4\right)}\parallel }_{\infty }.\end{array}$$

(4)

Let us recall some published papers about the above inequalities using QC that inspired us.

Regarding H-H type inequalities in QC, there are many recently published papers in this direction. For example, Tariboon and Ntouyas [35] obtained $\tilde{\mathrm{q}}$-analogue of trapezium’s inequality using the concepts of QC (also known as calculus without limits) on the finite intervals. By taking $\tilde{\mathrm{q}}\to 1^{-},$ we obtain classical calculus, see [36]. Alp et al. [37] obtained a corrected $\tilde{\mathrm{q}}$-analog of H-H inequality. Noor et al. [38] and Sudsutad et al. [39] derived some more $\tilde{\mathrm{q}}$-analogs of trapezium-like inequalities involving first-order $\tilde{\mathrm{q}}$-differentiable convex functions and Liu and Zhuang [40] established these analogs via second-order $\tilde{\mathrm{q}}$-differentiable convex functions. Budak et al. [41] obtain some refinements of quantum-type inequalities. Rashid et al. [42] derived new quantum integral inequalities for some new classes of generalized $\psi$-convex functions. Moreover, regarding Ostrowski and Simpson–Newton-like inequalities in QC, some good papers have been published. For example, Ali et al. [43] give some quantum Ostrowski-type inequalities for twice quantum differentiable functions and Butt et al. [44] derived new quantum Mercer estimates of Simpson–Newton-like inequalities via convexity.

Wang et al. [45] developed new Ostrowski-type inequalities via $\tilde{\mathrm{q}}$-fractional integrals involving s-convex functions. To the best of our knowledge, this is the first published paper working in this new direction that mixed together the FC with QC. Inspired from this paper, we attempt to give some new quantum fractional integral inequalities of trapezium, midpoint and Simpson’s type.

The following is the structure of this article: Section 2 provides a brief overview of the fundamentals FC and QC. Section 3 contains the main results of the article, which consists of proving a new parameterized quantum fractional integral identity that will be used to derive some integral inequalities of trapezium, midpoint and Simpson’s type via s-convex functions. From our main results, we discuss in detail several special cases. In Section 4, we derive an example and numerical analysis to show the efficiency of our theoretical results. Furthermore, numerical analysis will show that the mixed FC with QC give better estimates compared with FC or QC separately. This justifies the advantage of this proposed technique. In Section 5, we offer an application on special means of positive real numbers. In Section 6, conclusion and future research is given.

2. Preliminaries

Let us denote, respectively, $\mathcal{L}[{\vartheta }_1,{\vartheta }_2]$ the set of all Lebesgue integrable functions on $[{\vartheta }_1,{\vartheta }_2]$ and $\mathcal{C}[{\vartheta }_1,{\vartheta }_2]$ the set of all differentiable continuous functions on $[{\vartheta }_1,{\vartheta }_2].$

2.1. Fractional Calculus

Definition 3.

Let $\alpha >0,0\le {\vartheta }_1<{\vartheta }_2$ and $\mathrm{Y}\in \mathcal{L}[{\vartheta }_1,{\vartheta }_2].$ Then the R-L operators of order α are defined by

$${\mathrm{J}}_{{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\xi \right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\vartheta }_1}^{\xi }{(\xi -\varsigma )}^{\alpha -1}\mathrm{Y}\left(\varsigma \right)d\varsigma ,{\vartheta }_1<\xi$$

(5)

and

$${\mathrm{J}}_{{\vartheta }_2^{-}}^{\alpha }\mathrm{Y}\left(\xi \right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{\xi }^{{\vartheta }_2}{(\varsigma -\xi )}^{\alpha -1}\mathrm{Y}\left(\varsigma \right)d\varsigma ,\xi <{\vartheta }_2,$$

where $\mathsf{\Gamma }(·)$ is gamma function, defined by

$$\mathsf{\Gamma }\left(\alpha \right)={\int }_0^{\infty }{\varsigma }^{\alpha -1}{e}^{-\varsigma }d\varsigma ,\mathsf{\Gamma }(\alpha +1)=\alpha \mathsf{\Gamma }\left(\alpha \right).$$

For $\alpha =1,$ we obtain the classical Riemann integrals.

2.2. Quantum Calculus

Throughout the remaining paper, let $0<\tilde{\mathrm{q}}<1$ be a constant.

Definition 4

([36]). For $\mathrm{Y}\in \mathcal{C}[{\vartheta }_1,{\vartheta }_2],$ the left $\tilde{\mathrm{q}}$-derivative of Υ at $\xi \in [{\vartheta }_1,{\vartheta }_2]$ is given by

$$\begin{array}{c}\hfill {}_{{\vartheta }_1}{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\xi \right)=\frac{\mathrm{Y}\left(\xi \right)-\mathrm{Y}\left(\tilde{\mathrm{q}}\xi +(1-\tilde{\mathrm{q}}){\vartheta }_1\right)}{(1-\tilde{\mathrm{q}})(\xi -{\vartheta }_1)},\xi \ne {\vartheta }_1.\end{array}$$

(6)

The function Υ is said to be $\tilde{\mathrm{q}}$-differentiable on $[{\vartheta }_1,{\vartheta }_2]$ if ${}_{{\vartheta }_1}{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\xi \right)$ exists for all $\xi \in [{\vartheta }_1,{\vartheta }_2].$ If we choose ${\vartheta }_1=0,$ then we use the notation ${}_{{\vartheta }_1}{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\xi \right)={\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\xi \right),$ which is the $\tilde{\mathrm{q}}$-Jackson derivative [36,37,46].

The $\tilde{\mathrm{q}}$-integer is expressed as follows:

$$\begin{array}{c}\hfill {\left[n\right]}_{\tilde{\mathrm{q}}}:=\frac{{\tilde{\mathrm{q}}}^n-1}{\tilde{\mathrm{q}}-1}=1+\tilde{\mathrm{q}}+{\tilde{\mathrm{q}}}^2+\cdots +{\tilde{\mathrm{q}}}^{n-1},n\in \mathbb{N},\tilde{\mathrm{q}}\in (0,1).\end{array}$$

The following $\tilde{\mathrm{q}}$-integral along with its properties can be studied in [37].

Definition 5.

Suppose that $\mathrm{Y}\in \mathcal{C}[{\vartheta }_1,{\vartheta }_2].$ Then $\tilde{\mathrm{q}}$-definite integral for $\xi \in [{\vartheta }_1,{\vartheta }_2]$ is defined as

$$\begin{array}{c}\hfill {\int }_{{\vartheta }_1}^{\xi }\mathrm{Y}{\left(\varsigma \right)}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma =(1-\tilde{\mathrm{q}})(\xi -{\vartheta }_1)\sum _{r=0}^{\infty }{\tilde{\mathrm{q}}}^r\mathrm{Y}\left({\tilde{\mathrm{q}}}^r\xi +(1-{\tilde{\mathrm{q}}}^r){\vartheta }_1\right).\end{array}$$

(7)

Choosing ${\vartheta }_1=0$ in (7), we have

$$\begin{array}{c}\hfill {\int }_0^{\xi }\mathrm{Y}\left(\varsigma \right)d_{\tilde{\mathrm{q}}}\varsigma =(1-\tilde{\mathrm{q}})\xi \sum _{r=0}^{\infty }{\tilde{\mathrm{q}}}^r\mathrm{Y}\left({\tilde{\mathrm{q}}}^r\xi \right),\end{array}$$

which gives

$$\begin{array}{c}\hfill {\int }_0^1{\varsigma }^{\alpha +s}{d}_{\tilde{\mathrm{q}}}\varsigma =\frac{1}{{[\alpha +s+1]}_{\tilde{\mathrm{q}}}},{\int }_0^1{\varsigma }^{\alpha }{(1-\varsigma )}^s{d}_{\tilde{\mathrm{q}}}\varsigma =\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1){\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(s+1)}{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +s+2)},\end{array}$$

where $\tilde{\mathrm{q}}$-gamma function for $\xi >0$ is defined by

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}\left(\xi \right)={\int }_0^{\infty }{\varsigma }^{\xi -1}{\mathcal{E}}_{\tilde{\mathrm{q}}}^{-\tilde{\mathrm{q}}\varsigma }{d}_{\tilde{\mathrm{q}}}\varsigma ,{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\xi +1)={\left[\xi \right]}_{\tilde{\mathrm{q}}}{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}\left(\xi \right),\end{array}$$

and $\tilde{\mathrm{q}}$-exponential function is given as

$$\begin{array}{c}\hfill {\mathcal{E}}_{\tilde{\mathrm{q}}}^{\varsigma }=\sum _{r=0}^{\infty }{\tilde{\mathrm{q}}}^{\frac{r(r-1)}{2}}\frac{{\varsigma }^r}{{\left[r\right]}_{\tilde{\mathrm{q}}}!}.\end{array}$$

The following $\tilde{\mathrm{q}}$-fractional integrals can be studied in [47].

Definition 6.

Let $\alpha >0,0\le {\vartheta }_1<{\vartheta }_2$ and $\mathrm{Y}\in \mathcal{L}[{\vartheta }_1,{\vartheta }_2].$ Then, the $\tilde{\mathrm{q}}$-R-L of order α are defined by

$${\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\xi \right)=\frac{1}{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}\left(\alpha \right)}{\int }_{{\vartheta }_1}^{\xi }{(\xi -\tilde{\mathrm{q}}\varsigma )}^{\alpha -1}\mathrm{Y}\left(\varsigma \right){}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma ,{\vartheta }_1<\xi$$

(8)

and

$${\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_2^{-}}^{\alpha }\mathrm{Y}\left(\xi \right)=\frac{1}{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}\left(\alpha \right)}{\int }_{\tilde{\mathrm{q}}\xi }^{{\vartheta }_2}{(\varsigma -\tilde{\mathrm{q}}\xi )}^{\alpha -1}\mathrm{Y}\left(\varsigma \right)d_{\tilde{\mathrm{q}}}\varsigma ,\xi <{\vartheta }_2,$$

where ${\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(·)$ is $\tilde{\mathrm{q}}$-gamma function. For $\tilde{\mathrm{q}}\to 1^{-},$ we obtain R-L operators.

Theorem 4

((Formula for $\tilde{\mathrm{q}}$-integration by parts) [35]). Let ${\mathrm{Y}}_1,{\mathrm{Y}}_2\in \mathcal{C}[{\vartheta }_1,{\vartheta }_2],$ then for all $\xi \in [{\vartheta }_1,{\vartheta }_2],$ we have

$$\begin{array}{c}\hfill {\int }_{{\vartheta }_1}^{\xi }{\mathrm{Y}}_1\left(\varsigma \right){}_{{\vartheta }_1}{\mathcal{D}}_{\tilde{\mathrm{q}}}{\mathrm{Y}}_2\left(\varsigma \right){}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma ={\mathrm{Y}}_1\left(\xi \right){\mathrm{Y}}_2\left(\xi \right)-{\mathrm{Y}}_1\left({\vartheta }_1\right){\mathrm{Y}}_2\left({\vartheta }_1\right)\\ \hfill -{\int }_{{\vartheta }_1}^{\xi }{\mathrm{Y}}_2\left(\tilde{\mathrm{q}}\varsigma +(1-\tilde{\mathrm{q}}){\vartheta }_1\right){}_{{\vartheta }_1}{\mathcal{D}}_{\tilde{\mathrm{q}}}{\mathrm{Y}}_1\left(\varsigma \right){}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma .\end{array}$$

(9)

Theorem 5

(($\tilde{\mathrm{q}}$-Hölder’s inequality) [48]). Let ${\mathrm{Y}}_1,{\mathrm{Y}}_2$ be two $\tilde{\mathrm{q}}$-integrable functions on $[{\vartheta }_1,{\vartheta }_2]$ such that $p,{\tilde{\mathrm{q}}}_{*}>1,$ and $\frac{1}{p}+\frac{1}{{\tilde{\mathrm{q}}}_{*}}=1,$ then we have

$$\begin{array}{c}\hfill {\int }_{{\vartheta }_1}^{{\vartheta }_2}\left|{\mathrm{Y}}_1\left(\varsigma \right){\mathrm{Y}}_2\left(\varsigma \right)\right|{}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma \le {\left({\int }_{{\vartheta }_1}^{{\vartheta }_2}{\left|{\mathrm{Y}}_1\left(\varsigma \right)\right|}^p{}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{\frac{1}{p}}{\left({\int }_{{\vartheta }_1}^{{\vartheta }_2}{\left|{\mathrm{Y}}_2\left(\varsigma \right)\right|}^{{\tilde{\mathrm{q}}}_{*}}{}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}.\end{array}$$

(10)

Theorem 6

(($\tilde{\mathrm{q}}$-Power mean inequality) [48]). Let ${\mathrm{Y}}_1,{\mathrm{Y}}_2$ be two $\tilde{\mathrm{q}}$-integrable functions on $[{\vartheta }_1,{\vartheta }_2]$ such that ${\tilde{\mathrm{q}}}_{*}\ge 1,$ then we have

$$\begin{array}{c}\hfill {\int }_{{\vartheta }_1}^{{\vartheta }_2}\left|{\mathrm{Y}}_1\left(\varsigma \right){\mathrm{Y}}_2\left(\varsigma \right)\right|{}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma \le {\left({\int }_{{\vartheta }_1}^{{\vartheta }_2}\left|{\mathrm{Y}}_1\left(\varsigma \right)\right|{}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}{\left({\int }_{{\vartheta }_1}^{{\vartheta }_2}\left|{\mathrm{Y}}_1\left(\varsigma \right)\right|{\left|{\mathrm{Y}}_2\left(\varsigma \right)\right|}^{{\tilde{\mathrm{q}}}_{*}}{}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}.\end{array}$$

(11)

3. Main Results

For the simplicity of notation, let

$$\delta (\xi ,\alpha ):={\int }_0^1\left|\xi -{\varsigma }^{\alpha }\right|d\varsigma ,\rho (\xi ,p,\alpha ):={\int }_0^1{\left|\xi -{\varsigma }^{\alpha }\right|}^{p}d\varsigma .$$

Let us recall the well-known beta and hypergeometric functions below:

$$\beta (x,y):={\int }_0^1{\varsigma }^{x-1}{(1-\varsigma )}^{y-1}d\varsigma ,x,y>0$$

and

$${}_2{\mathcal{F}}_1\left({\vartheta }_1,{\vartheta }_2;{\vartheta }_3;z\right):=\frac{1}{\beta ({\vartheta }_2,{\vartheta }_3-{\vartheta }_2)}{\int }_0^1{\varsigma }^{{\vartheta }_2-1}{(1-\varsigma )}^{{\vartheta }_3-{\vartheta }_2-1}{(1-z\varsigma )}^{-{\vartheta }_1}d\varsigma ,\mathfrak{R}\left({\vartheta }_3\right)>\mathfrak{R}\left({\vartheta }_2\right)>0,\left|z\right|\le 1.$$

The following two lemmas are very useful in the sequel.

Lemma 1.

For $\alpha >0$ and $0\le \xi \le 1,$ we have

$$\delta (\xi ,\alpha ):=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\alpha +1},}\hfill & \xi =0;\hfill \\ \\ {\displaystyle \frac{2\alpha {\xi }^{1+\frac{1}{\alpha }}+1}{\alpha +1}-\xi ,}\hfill & 0<\xi <1;\hfill \\ \\ {\displaystyle \frac{\alpha }{\alpha +1},}\hfill & \xi =1.\hfill \end{array}\right.$$

Proof. 

The proof is evident. □

Lemma 2.

For $\alpha >0,p\ge 1$ and $0\le \xi \le 1,$ we have

$$\rho (\xi ,p,\alpha ):=\left\{\begin{array}{cc}{\displaystyle \frac{1}{p\alpha +1},}\hfill & \xi =0;\hfill \\ \\ {\displaystyle \frac{{\xi }^{p+\frac{1}{\alpha }}}{\alpha }\beta \left(\frac{1}{\alpha },p+1\right)+\frac{{(1-\xi )}^{p+1}}{\alpha (p+1)}}\hfill \\ {\displaystyle \times {}_2{\mathcal{F}}_1\left(1-\frac{1}{\alpha },1;p+2;1-\xi \right),}\hfill & 0<\xi <1;\hfill \\ \\ {\displaystyle \frac{1}{\alpha }\beta \left(\frac{1}{\alpha },p+1\right),}\hfill & \xi =1.\hfill \end{array}\right.$$

Proof. 

The proof is a straightforward computation; we omit here their details. □

We are in position to prove a new lemma including $\tilde{\mathrm{q}}$-R-L operators in order to establish our main results.

Lemma 3.

Let $\mathrm{Y}:[{\vartheta }_1,{\vartheta }_2]\to \mathbb{R}$ be a $\tilde{\mathrm{q}}$-differentiable function, where $0<\tilde{\mathrm{q}}<1$ such that $0\le {\vartheta }_1<{\vartheta }_2$. If ${\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\in \mathcal{L}[{\vartheta }_1,{\vartheta }_2],\eta ,\sigma \in \mathbb{R}$ and $\alpha \in \mathbb{N},$ then we have

$$\begin{array}{cc}\hfill & \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]\hfill \\ \hfill & =\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\hfill \\ \hfill & \times {\int }_0^1\left[\left(1-\eta -{\varsigma }^{\alpha }\right){\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\left(\sigma -{\varsigma }^{\alpha }\right){\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)\right]d_{\tilde{\mathrm{q}}}\varsigma .\hfill \end{array}$$

(12)

Proof. 

Let us denote, respectively,

$${\mathrm{I}}_1:={\int }_0^1\left(1-\eta -{\varsigma }^{\alpha }\right){\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)d_{\tilde{\mathrm{q}}}\varsigma$$

and

$${\mathrm{I}}_2:={\int }_0^1\left(\sigma -{\varsigma }^{\alpha }\right){\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)d_{\tilde{\mathrm{q}}}\varsigma .$$

By using the formula of $\tilde{\mathrm{q}}$-integration by parts, we have

$$\begin{array}{cc}\hfill & {\mathrm{I}}_1=(1-\eta ){\int }_0^1{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)d_{\tilde{\mathrm{q}}}\varsigma -{\int }_0^1{\varsigma }^{\alpha }{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)d_{\tilde{\mathrm{q}}}\varsigma \hfill \\ \hfill & =\frac{2(1-\eta )}{{\vartheta }_2-{\vartheta }_1}{\int }_{{\vartheta }_1}^{\frac{{\vartheta }_1+{\vartheta }_2}{2}}{}_{{\vartheta }_1}{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \right){}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma -{\varsigma }^{\alpha }\frac{\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}{{\vartheta }_1-\frac{{\vartheta }_1+{\vartheta }_2}{2}}{|}_0^1\hfill \\ \hfill & +\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}}{{\vartheta }_1-\frac{{\vartheta }_1+{\vartheta }_2}{2}}{\int }_0^1{\varsigma }^{\alpha -1}\mathrm{Y}\left(\tilde{\mathrm{q}}\varsigma {\vartheta }_1+(1-\tilde{\mathrm{q}}\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)d_{\tilde{\mathrm{q}}}\varsigma \hfill \\ \hfill & =\frac{2(1-\eta )}{{\vartheta }_2-{\vartheta }_1}\left(\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)-\mathrm{Y}\left({\vartheta }_1\right)\right)-\frac{2}{{\vartheta }_1-{\vartheta }_2}\mathrm{Y}\left({\vartheta }_1\right)+\frac{2{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\mathrm{Y}\left({\vartheta }_1\right)\hfill \\ \hfill & -\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}\left(\alpha \right)}{{\tilde{\mathrm{q}}}^{\alpha }}{\left(\frac{2}{{\vartheta }_2-{\vartheta }_1}\right)}^{\alpha +1}\frac{1}{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}\left(\alpha \right)}{\int }_{{\vartheta }_1}^{\frac{{\vartheta }_1+{\vartheta }_2}{2}}{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}-\tilde{\mathrm{q}}\varsigma \right)}^{\alpha -1}\mathrm{Y}\left(\varsigma \right)d_{\tilde{\mathrm{q}}}\varsigma \hfill \\ \hfill & =\frac{2(1-\eta )}{{\vartheta }_2-{\vartheta }_1}\left(\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)-\mathrm{Y}\left({\vartheta }_1\right)\right)+\frac{2}{{\vartheta }_2-{\vartheta }_1}\mathrm{Y}\left({\vartheta }_1\right)+\frac{2{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\mathrm{Y}\left({\vartheta }_1\right)\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }}{\left(\frac{2}{{\vartheta }_2-{\vartheta }_1}\right)}^{\alpha +1}{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right).\hfill \end{array}$$

(13)

Similarly, we obtain

$$\begin{array}{cc}\hfill & {\mathrm{I}}_2=\sigma {\int }_0^1{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)d_{\tilde{\mathrm{q}}}\varsigma -{\int }_0^1{\varsigma }^{\alpha }{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)d_{\tilde{\mathrm{q}}}\varsigma \hfill \\ \hfill & =\frac{2\sigma }{{\vartheta }_2-{\vartheta }_1}\left(\mathrm{Y}\left({\vartheta }_2\right)-\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)+\frac{2}{{\vartheta }_2-{\vartheta }_1}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\frac{2{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }}{\left(\frac{2}{{\vartheta }_2-{\vartheta }_1}\right)}^{\alpha +1}{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right).\hfill \end{array}$$

(14)

By adding (13) and (14) and multiplying them by ${\displaystyle \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}},$ we obtain the desired identity (12). □

Remark 1.

Taking $\tilde{\mathrm{q}}\to 1^{-}$ in Lemma 3, we have the following fractional identity:

$$\begin{array}{cc}\hfill & \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left(\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)\right)\hfill \\ \hfill & -\frac{\mathsf{\Gamma }(\alpha +1)}{{\vartheta }_2-{\vartheta }_1}\left[{\mathrm{J}}_{{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]\hfill \\ \hfill & =\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}{\int }_0^1\left[\left(1-\eta -{\varsigma }^{\alpha }\right){\mathrm{Y}}^{\prime }\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\left(\sigma -{\varsigma }^{\alpha }\right){\mathrm{Y}}^{\prime }\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)\right]d\varsigma .\hfill \end{array}$$

(15)

Remark 2.

Choosing $\alpha =1$ in Lemma 3, we obtain the following $\tilde{\mathrm{q}}$-identity:

$$\begin{array}{cc}\hfill & \frac{1}{2}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{1-\tilde{\mathrm{q}}}{\tilde{\mathrm{q}}}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{1}{\tilde{\mathrm{q}}({\vartheta }_2-{\vartheta }_1)}{\int }_{{\vartheta }_1}^{{\vartheta }_2}\mathrm{Y}\left(\varsigma \right){}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma \hfill \\ \hfill & =\frac{({\vartheta }_2-{\vartheta }_1)}{4}{\int }_0^1\left[\left(1-\eta -\varsigma \right){\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\left(\sigma -\varsigma \right){\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)\right]d_{\tilde{\mathrm{q}}}\varsigma .\hfill \end{array}$$

(16)

By using Lemmas 1–3, we established the following $\tilde{\mathrm{q}}$-fractional integral inequalities.

Theorem 7.

Let $\mathrm{Y}:[{\vartheta }_1,{\vartheta }_2]\to \mathbb{R}$ be a $\tilde{\mathrm{q}}$-differentiable function, where $0<\tilde{\mathrm{q}}<1$ such that $0\le {\vartheta }_1<{\vartheta }_2$ and $\eta ,\sigma \in [0,1]$. If ${\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\in \mathcal{L}[{\vartheta }_1,{\vartheta }_2]$ and $|{\mathcal{D}}_{\tilde{\mathrm{q}}}{\mathrm{Y}|}^{{\tilde{\mathrm{q}}}_{*}}$ is s-convex function for some fixed $s\in (0,1]$ such that $p,{\tilde{\mathrm{q}}}_{*}>1,$ and $\frac{1}{p}+\frac{1}{{\tilde{\mathrm{q}}}_{*}}=1,$ then for $\alpha \in \mathbb{N},$ the following $\tilde{\mathrm{q}}$-fractional integral inequality holds true:

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}{[s+1]}_{\tilde{\mathrm{q}}}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\{{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(\eta ,p,\alpha ){\left[|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(1-\sigma ,p,\alpha ){\left[{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\},\hfill \end{array}$$

(17)

where

$${\mathrm{A}}_{\tilde{\mathrm{q}}}(\eta ,p,\alpha ):={\int }_0^1{|1-\eta -{\varsigma }^{\alpha }|}^p{d}_{\tilde{\mathrm{q}}}\varsigma .$$

Proof. 

By using Lemma 3, $\tilde{\mathrm{q}}$-Hölder’s inequality, s-convexity of $|{\mathcal{D}}_{\tilde{\mathrm{q}}}{\mathrm{Y}|}^{{\tilde{\mathrm{q}}}_{*}}$ and properties of modulus, we have

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\int }_0^1|1-\eta -{\varsigma }^{\alpha }|\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|d_{\tilde{\mathrm{q}}}\varsigma \hfill \\ \hfill & +{\int }_0^1|\sigma -{\varsigma }^{\alpha }|\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)\right|d_{\tilde{\mathrm{q}}}\varsigma \}\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\left({\int }_0^1{|1-\eta -{\varsigma }^{\alpha }|}^p{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{\frac{1}{p}}{\left({\int }_0^1{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\left({\int }_0^1{|\sigma -{\varsigma }^{\alpha }|}^p{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{\frac{1}{p}}{\left({\int }_0^1{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(\eta ,p,\alpha ){\left[{\int }_0^1\left({\varsigma }^s{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{(1-\varsigma )}^s{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right)d_{\tilde{\mathrm{q}}}\varsigma \right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(1-\sigma ,p,\alpha ){\left[{\int }_0^1\left({\varsigma }^s{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{(1-\varsigma )}^s{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right)d_{\tilde{\mathrm{q}}}\varsigma \right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}\hfill \\ \hfill & =\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}{[s+1]}_{\tilde{\mathrm{q}}}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\{{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(\eta ,p,\alpha ){\left[|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(1-\sigma ,p,\alpha ){\left[{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}.\hfill \end{array}$$

The proof of Theorem 7 is completed. □

Corollary 1.

Taking $\tilde{\mathrm{q}}\to 1^{-}$ in Theorem 7, we have

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left(\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)\right)\hfill \\ \hfill & -\frac{\mathsf{\Gamma }(\alpha +1)}{{\vartheta }_2-{\vartheta }_1}\left[{\mathrm{J}}_{{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}{(s+1)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\{{\rho }^{\frac{1}{p}}(1-\eta ,p,\alpha ){\left[|{\mathrm{Y}}^{\prime }\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathrm{Y}}^{\prime }\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\rho }^{\frac{1}{p}}(\sigma ,p,\alpha ){\left[{\left|{\mathrm{Y}}^{\prime }\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathrm{Y}}^{\prime }\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}.\hfill \end{array}$$

(18)

Corollary 2.

Choosing $\alpha =1$ in Theorem 7, we obtain

$$\begin{array}{cc}\hfill & |\frac{1}{2}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{1-\tilde{\mathrm{q}}}{\tilde{\mathrm{q}}}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{1}{\tilde{\mathrm{q}}({\vartheta }_2-{\vartheta }_1)}{\int }_{{\vartheta }_1}^{{\vartheta }_2}\mathrm{Y}\left(\varsigma \right){}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma |\hfill \\ \hfill & \le \frac{({\vartheta }_2-{\vartheta }_1)}{4{[s+1]}_{\tilde{\mathrm{q}}}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\{{\mathrm{R}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(\eta ,p){\left[|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{R}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(1-\sigma ,p){\left[{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\},\hfill \end{array}$$

(19)

where

$${\mathrm{R}}_{\tilde{\mathrm{q}}}(\eta ,p):={\int }_0^1{|1-\eta -\varsigma |}^p{d}_{\tilde{\mathrm{q}}}\varsigma .$$

Corollary 3.

Taking $|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}|\le \mathcal{K}$ in Theorem 7, we obtain

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{\mathcal{K}{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha -\frac{1}{{\tilde{\mathrm{q}}}_{*}}+1}{[s+1]}_{\tilde{\mathrm{q}}}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\left({\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(\eta ,p,\alpha )+{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(1-\sigma ,p,\alpha )\right).\hfill \end{array}$$

(20)

Corollary 4.

Choosing $s=1$ in Theorem 7, we have

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}{(1+\tilde{\mathrm{q}})}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\{{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(\eta ,p,\alpha ){\left[|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(1-\sigma ,p,\alpha ){\left[{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}.\hfill \end{array}$$

(21)

Corollary 5.

Taking $\eta =\sigma$ in Theorem 7, we obtain

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left({\vartheta }_2\right)\right)+2(1-\eta )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}{[s+1]}_{\tilde{\mathrm{q}}}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\{{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(\eta ,p,\alpha ){\left[|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(1-\eta ,p,\alpha ){\left[{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}.\hfill \end{array}$$

(22)

Corollary 6.

Choosing $\eta =\frac{1}{2},\frac{2}{3},\frac{1}{3},$ respectively, in Corollary 5, we obtain the following $\tilde{\mathrm{q}}$-fractional integral inequalities of trapezium, midpoint and Simpson’s type:

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\frac{\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left({\vartheta }_2\right)}{2}+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}{[s+1]}_{\tilde{\mathrm{q}}}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}\left(\frac{1}{2},p,\alpha \right)\{{\left[|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\left[{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\};\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\frac{2}{3}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\mathrm{Y}\left({\vartheta }_2\right)\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}{[s+1]}_{\tilde{\mathrm{q}}}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\{{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}\left(\frac{2}{3},p,\alpha \right){\left[|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}\left(\frac{1}{3},p,\alpha \right){\left[{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\};\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\frac{1}{3}\left(\mathrm{Y}\left({\vartheta }_1\right)+4\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\mathrm{Y}\left({\vartheta }_2\right)\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}{[s+1]}_{\tilde{\mathrm{q}}}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\{{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}\left(\frac{1}{3},p,\alpha \right){\left[|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{A}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}\left(\frac{2}{3},p,\alpha \right){\left[{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}.\hfill \end{array}$$

(25)

Theorem 8.

Let $\mathrm{Y}:[{\vartheta }_1,{\vartheta }_2]\to \mathbb{R}$ be a $\tilde{\mathrm{q}}$-differentiable function, where $0<\tilde{\mathrm{q}}<1$ such that $0\le {\vartheta }_1<{\vartheta }_2$ and $\eta ,\sigma \in [0,1]$. If ${\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\in \mathcal{L}[{\vartheta }_1,{\vartheta }_2]$ and $|{\mathcal{D}}_{\tilde{\mathrm{q}}}{\mathrm{Y}|}^{{\tilde{\mathrm{q}}}_{*}}$ is s-convex function for some fixed $s\in (0,1]$ such that ${\tilde{\mathrm{q}}}_{*}\ge 1,$ then for $\alpha \in \mathbb{N},$ the following $\tilde{\mathrm{q}}$-fractional integral inequality holds true:

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(\eta ,\alpha ){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(1-\sigma ,\alpha ){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(1-\sigma ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}(1-\sigma ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\},\hfill \end{array}$$

(26)

where

$${\mathrm{M}}_{\tilde{\mathrm{q}}}(\eta ,\alpha ):={\int }_0^1|1-\eta -{\varsigma }^{\alpha }|d_{\tilde{\mathrm{q}}}\varsigma$$

and

$${\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha ):={\int }_0^1{\varsigma }^s|1-\eta -{\varsigma }^{\alpha }|d_{\tilde{\mathrm{q}}}\varsigma ,{\mathrm{E}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha ):={\int }_0^1{(1-\varsigma )}^s|1-\eta -{\varsigma }^{\alpha }|d_{\tilde{\mathrm{q}}}\varsigma .$$

Proof. 

By using Lemma 3, $\tilde{\mathrm{q}}$-Power mean inequality, s-convexity of $|{\mathcal{D}}_{\tilde{\mathrm{q}}}{\mathrm{Y}|}^{{\tilde{\mathrm{q}}}_{*}}$ and properties of modulus, we have

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\int }_0^1|1-\eta -{\varsigma }^{\alpha }|\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|d_{\tilde{\mathrm{q}}}\varsigma \hfill \\ \hfill & +{\int }_0^1|\sigma -{\varsigma }^{\alpha }|\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)\right|d_{\tilde{\mathrm{q}}}\varsigma \}\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\left({\int }_0^1|1-\eta -{\varsigma }^{\alpha }|d_{\tilde{\mathrm{q}}}\varsigma \right)}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}{\left({\int }_0^1|1-\eta -{\varsigma }^{\alpha }|{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma {\vartheta }_1+(1-\varsigma )\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\left({\int }_0^1|\sigma -{\varsigma }^{\alpha }|d_{\tilde{\mathrm{q}}}\varsigma \right)}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}{\left({\int }_0^1|\sigma -{\varsigma }^{\alpha }|{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \frac{{\vartheta }_1+{\vartheta }_2}{2}+(1-\varsigma ){\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}{d}_{\tilde{\mathrm{q}}}\varsigma \right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(\eta ,\alpha ){\left[{\int }_0^1|1-\eta -{\varsigma }^{\alpha }|\left({\varsigma }^s{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{(1-\varsigma )}^s{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right)d_{\tilde{\mathrm{q}}}\varsigma \right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(1-\sigma ,\alpha ){\left[{\int }_0^1|\sigma -{\varsigma }^{\alpha }|\left({\varsigma }^s{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{(1-\varsigma )}^s{\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right)d_{\tilde{\mathrm{q}}}\varsigma \right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}\hfill \\ \hfill & =\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(\eta ,\alpha ){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(1-\sigma ,\alpha ){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(1-\sigma ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}(1-\sigma ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\},\hfill \end{array}$$

which completes the proof of Theorem 8. □

Corollary 7.

Taking ${\tilde{\mathrm{q}}}_{*}=1$ in Theorem 8, we have

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\hfill \\ \hfill & \times \left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha )\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|+\left({\mathrm{E}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha )+{\mathrm{C}}_{\tilde{\mathrm{q}}}(1-\sigma ,s,\alpha )\right)\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|+{\mathrm{E}}_{\tilde{\mathrm{q}}}(1-\sigma ,s,\alpha )\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|\right].\hfill \end{array}$$

(27)

Corollary 8.

Choosing $\tilde{\mathrm{q}}\to 1^{-}$ in Theorem 8, we obtain

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left(\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)\right)\hfill \\ \hfill & -\frac{\mathsf{\Gamma }(\alpha +1)}{{\vartheta }_2-{\vartheta }_1}\left[{\mathrm{J}}_{{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\delta }^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(1-\eta ,\alpha ){\left[\mathrm{C}(\eta ,s,\alpha )|{\mathrm{Y}}^{\prime }\left({\vartheta }_1\right){|}^{{\tilde{\mathrm{q}}}_{*}}+\mathrm{E}(\eta ,s,\alpha ){\left|{\mathrm{Y}}^{\prime }\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\delta }^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(\sigma ,\alpha ){\left[\mathrm{C}(1-\sigma ,s,\alpha ){\left|{\mathrm{Y}}^{\prime }\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+\mathrm{E}(1-\sigma ,s,\alpha ){\left|{\mathrm{Y}}^{\prime }\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\},\hfill \end{array}$$

(28)

where

$$\mathrm{C}(\eta ,s,\alpha ):={\int }_0^1{\varsigma }^s|1-\eta -{\varsigma }^{\alpha }|d\varsigma ,\mathrm{E}(\eta ,s,\alpha ):={\int }_0^1{(1-\varsigma )}^s|1-\eta -{\varsigma }^{\alpha }|d\varsigma .$$

Corollary 9.

Taking $\alpha =1$ in Theorem 8, we obtain

$$\begin{array}{cc}\hfill & |\frac{1}{2}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{1-\tilde{\mathrm{q}}}{\tilde{\mathrm{q}}}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{1}{\tilde{\mathrm{q}}({\vartheta }_2-{\vartheta }_1)}{\int }_{{\vartheta }_1}^{{\vartheta }_2}\mathrm{Y}\left(\varsigma \right){}_{{\vartheta }_1}{d}_{\tilde{\mathrm{q}}}\varsigma |\hfill \\ \hfill & \le \frac{({\vartheta }_2-{\vartheta }_1)}{4}\{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\eta \right){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,s){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}(\eta ,s){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(1-\sigma ){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(1-\sigma ,s){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}(1-\sigma ,s){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\},\hfill \end{array}$$

(29)

where

$${\mathrm{M}}_{\tilde{\mathrm{q}}}\left(\eta \right):={\int }_0^1|1-\eta -\varsigma |d_{\tilde{\mathrm{q}}}\varsigma$$

and

$${\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,s):={\int }_0^1{\varsigma }^s|1-\eta -\varsigma |d_{\tilde{\mathrm{q}}}\varsigma ,{\mathrm{E}}_{\tilde{\mathrm{q}}}(\eta ,s):={\int }_0^1{(1-\varsigma )}^s|1-\eta -\varsigma |d_{\tilde{\mathrm{q}}}\varsigma .$$

Corollary 10.

Choosing $|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}|\le \mathcal{K}$ in Theorem 8, we have

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{\mathcal{K}{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(\eta ,\alpha ){\left({\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha )+{\mathrm{E}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha )\right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(1-\sigma ,\alpha ){\left({\mathrm{C}}_{\tilde{\mathrm{q}}}(1-\sigma ,s,\alpha )+{\mathrm{E}}_{\tilde{\mathrm{q}}}(1-\sigma ,s,\alpha )\right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}.\hfill \end{array}$$

(30)

Corollary 11.

Taking $s=1$ in Theorem 8, we obtain

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \mathrm{Y}\left({\vartheta }_1\right)+(2-\eta -\sigma )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\sigma \mathrm{Y}\left({\vartheta }_2\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(\eta ,\alpha ){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+\left({\mathrm{M}}_{\tilde{\mathrm{q}}}(\eta ,\alpha )-{\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,\alpha )\right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(1-\sigma ,\alpha ){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(1-\sigma ,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+\left({\mathrm{M}}_{\tilde{\mathrm{q}}}(1-\sigma ,\alpha )-{\mathrm{C}}_{\tilde{\mathrm{q}}}(1-\sigma ,\alpha )\right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\},\hfill \end{array}$$

(31)

where

$${\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,\alpha ):={\int }_0^1\varsigma |1-\eta -{\varsigma }^{\alpha }|d_{\tilde{\mathrm{q}}}\varsigma .$$

Corollary 12.

Choosing $\eta =\sigma$ in Theorem 8, we obtain

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\eta \left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left({\vartheta }_2\right)\right)+2(1-\eta )\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(\eta ,\alpha ){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}(\eta ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(1-\eta ,\alpha ){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}(1-\eta ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}(1-\eta ,s,\alpha ){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}.\hfill \end{array}$$

(32)

Corollary 13.

Taking $\eta =\frac{1}{2},\frac{2}{3},\frac{1}{3},$ respectively, in Corollary 12, we have the following $\tilde{\mathrm{q}}$-fractional integral inequalities of trapezium, midpoint and Simpson’s type:

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\frac{\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left({\vartheta }_2\right)}{2}+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\frac{1}{2},\alpha \right)\{{\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{1}{2},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{1}{2},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{1}{2},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{1}{2},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\};\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\frac{2}{3}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\mathrm{Y}\left({\vartheta }_2\right)\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\frac{2}{3},\alpha \right){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{2}{3},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{2}{3},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\frac{1}{3},\alpha \right){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{1}{3},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{1}{3},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\};\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & |\frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha -1}}{2^{\alpha }}\left[\frac{1}{3}\left(\mathrm{Y}\left({\vartheta }_1\right)+4\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+\mathrm{Y}\left({\vartheta }_2\right)\right)+\frac{{\left[\alpha \right]}_{\tilde{\mathrm{q}}}(1-\tilde{\mathrm{q}})}{{\tilde{\mathrm{q}}}^{\alpha }}\left(\mathrm{Y}\left({\vartheta }_1\right)+\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathsf{\Gamma }}_{\tilde{\mathrm{q}}}(\alpha +1)}{{\tilde{\mathrm{q}}}^{\alpha }({\vartheta }_2-{\vartheta }_1)}\left[{\mathrm{J}}_{\tilde{\mathrm{q}},{\vartheta }_1^{+}}^{\alpha }\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)+{\mathrm{J}}_{\tilde{\mathrm{q}},{\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)}^{+}}^{\alpha }\mathrm{Y}\left({\vartheta }_2\right)\right]|\hfill \\ \hfill & \le \frac{{({\vartheta }_2-{\vartheta }_1)}^{\alpha }}{2^{\alpha +1}}\{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\frac{1}{3},\alpha \right){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{1}{3},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_1\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{1}{3},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & + {\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\frac{2}{3},\alpha \right){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{2}{3},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\frac{{\vartheta }_1+{\vartheta }_2}{2}\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{2}{3},s,\alpha \right){\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left({\vartheta }_2\right)\right|}^{{\tilde{\mathrm{q}}}_{*}}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}.\hfill \end{array}$$

(35)

4. Example and Numerical Analysis

4.1. Example

Let $\mathrm{Y}\left(\varsigma \right)=\frac{{\varsigma }^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}}{{\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1\right]}_{\tilde{\mathrm{q}}}},$ where $n\in \mathbb{N},{\tilde{\mathrm{q}}}_{*}\ge 1$ and $\tilde{\mathrm{q}}\in (0,1).$ After simple calculations, we have ${\left|{\mathcal{D}}_{\tilde{\mathrm{q}}}\mathrm{Y}\left(\varsigma \right)\right|}^{{\tilde{\mathrm{q}}}_{*}}={\varsigma }^{n+1},$ which shows that $|{\mathcal{D}}_{\tilde{\mathrm{q}}}{\mathrm{Y}\left(\varsigma \right)|}^{{\tilde{\mathrm{q}}}_{*}}$ is convex function for all $\varsigma >0$ and $n\in \mathbb{N}.$ Then, by applying Corollaries 2 and 9 for specific values $s=1,{\vartheta }_1=0,{\vartheta }_2=1,$ and $\eta ,\sigma \in [0,1],$ we deduce the following $\tilde{\mathrm{q}}$-inequalities:

$$\begin{array}{cc}\hfill & |\frac{1}{2}\left[(2-\eta -\sigma ){\left(\frac{1}{2}\right)}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}+\sigma +\frac{1-\tilde{\mathrm{q}}}{\tilde{\mathrm{q}}}{\left(\frac{1}{2}\right)}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\right]-\frac{1}{\tilde{\mathrm{q}}{\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+2\right]}_{\tilde{\mathrm{q}}}}|\le \frac{{\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1\right]}_{\tilde{\mathrm{q}}}}{4{(1+\tilde{\mathrm{q}})}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\hfill \\ \hfill & \times \left\{{\mathrm{R}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(\eta ,p){\left(\frac{1}{2}\right)}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}}+{\mathrm{R}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}(1-\sigma ,p){\left[1+{\left(\frac{1}{2}\right)}^{n+1}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\right\};\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & |\frac{1}{2}\left[(2-\eta -\sigma ){\left(\frac{1}{2}\right)}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}+\sigma +\frac{1-\tilde{\mathrm{q}}}{\tilde{\mathrm{q}}}{\left(\frac{1}{2}\right)}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\right]-\frac{1}{\tilde{\mathrm{q}}{\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+2\right]}_{\tilde{\mathrm{q}}}}|\le \frac{{\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1\right]}_{\tilde{\mathrm{q}}}}{4}\hfill \\ \hfill & \times \{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\eta \right){\left[{\left(\frac{1}{2}\right)}^{n+1}\left({\mathrm{M}}_{\tilde{\mathrm{q}}}\left(\eta \right)-{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\eta \right)\right)\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & +{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}(1-\sigma ){\left[\left({\left(\frac{1}{2}\right)}^{n+1}-1\right){\mathrm{C}}_{\tilde{\mathrm{q}}}(1-\sigma )+{\mathrm{M}}_{\tilde{\mathrm{q}}}(1-\sigma )\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\},\hfill \end{array}$$

(37)

where

$${\mathrm{M}}_{\tilde{\mathrm{q}}}\left(\eta \right):={\int }_0^1|1-\eta -\varsigma |d_{\tilde{\mathrm{q}}}\varsigma ,{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\eta \right):={\int }_0^1\varsigma |1-\eta -\varsigma |d_{\tilde{\mathrm{q}}}\varsigma .$$

4.2. Numerical Analysis

Let us choose ${\vartheta }_1=0,{\vartheta }_2=1$ with $s=n={\tilde{\mathrm{q}}}_{*}=1,\alpha \in \mathbb{N}$ and $\tilde{\mathrm{q}}\in (0,1).$ Using the function $\mathrm{Y}\left(\varsigma \right)=\frac{{\varsigma }^3}{{\left[3\right]}_{\tilde{\mathrm{q}}}}$ in Theorem 8, the following

Table 1. Comparison results in Theorem 8 between L.H.S and R.H.S using FC, QC and mixed FC with QC.

Values of $\mathit{\eta },\mathit{\sigma }$	FC	QC	Mixed FC with QC
$(\eta ,\sigma )$	$\alpha =2,\tilde{\mathrm{q}}\to 1^{-}$	$\alpha =1,\tilde{\mathrm{q}}={\displaystyle \frac{1}{2}}$	$\alpha =2,\tilde{\mathrm{q}}={\displaystyle \frac{1}{2}}$
(0, 0)	$0.0208\le 0.0312$	$0.1302\le 0.8791$	$0.0306\le 0.1004$
(1, 1)	$0.0416\le 0.0625$	$0.1250\le 0.5041$	$0.0422\le 0.4355$

shows that the mixed FC with QC give better estimates compared to FC or QC separately.

5. Application to Special Means

We consider the following arithmetic mean of real numbers ${\vartheta }_1$ and ${\vartheta }_2$ such that $0\le {\vartheta }_1<{\vartheta }_2$:

$$\mathcal{A}({\vartheta }_1,{\vartheta }_2)=\frac{{\vartheta }_1+{\vartheta }_2}{2}.$$

For the simplicity of notation, let

$${\Delta }_{\tilde{\mathrm{q}}}\left(n,{\tilde{\mathrm{q}}}_{*};{\vartheta }_1,{\vartheta }_2\right):=(1-\tilde{\mathrm{q}})({\vartheta }_2-{\vartheta }_1)\sum _{r=0}^{\infty }{\tilde{\mathrm{q}}}^r{\left({\tilde{\mathrm{q}}}^r{\vartheta }_2+(1-{\tilde{\mathrm{q}}}^r){\vartheta }_1\right)}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1},$$

where $n\in \mathbb{N},{\tilde{\mathrm{q}}}_{*}\ge 1,$ and $\tilde{\mathrm{q}}\in (0,1).$

Proposition 1.

Let $n\in \mathbb{N},\tilde{\mathrm{q}}\in (0,1)$ and ${\vartheta }_1,{\vartheta }_2\in \mathbb{R},$ where $0\le {\vartheta }_1<{\vartheta }_2.$ Then, for $p,{\tilde{\mathrm{q}}}_{*}>1$ and $\frac{1}{p}+\frac{1}{{\tilde{\mathrm{q}}}_{*}}=1,$ we have

$$\begin{array}{cc}\hfill & |\mathcal{A}\left({\vartheta }_1^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1},{\vartheta }_2^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\right)+{\mathcal{A}}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\left({\vartheta }_1,{\vartheta }_2\right)+\frac{2(1-\tilde{\mathrm{q}})}{\tilde{\mathrm{q}}}\mathcal{A}\left({\vartheta }_1^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1},{\mathcal{A}}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\left({\vartheta }_1,{\vartheta }_2\right)\right)-\frac{2{\Delta }_{\tilde{\mathrm{q}}}\left(n,{\tilde{\mathrm{q}}}_{*};{\vartheta }_1,{\vartheta }_2\right)}{\tilde{\mathrm{q}}({\vartheta }_2-{\vartheta }_1)}|\hfill \\ \hfill & \le \frac{({\vartheta }_2-{\vartheta }_1){\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1\right]}_{\tilde{\mathrm{q}}}}{2{(1+\tilde{\mathrm{q}})}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\hfill \\ \hfill & \times {\mathrm{R}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}\left(\frac{1}{2},p\right)\left\{{\left({\vartheta }_1^{n+1}+{\mathcal{A}}^{n+1}({\vartheta }_1,{\vartheta }_2)\right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}+{\left({\mathcal{A}}^{n+1}({\vartheta }_1,{\vartheta }_2)+{\vartheta }_2^{n+1}\right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\right\},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & |\frac{2}{3}\left[\mathcal{A}\left({\vartheta }_1^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1},{\vartheta }_2^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\right)+2{\mathcal{A}}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\left({\vartheta }_1,{\vartheta }_2\right)\right]+\frac{2(1-\tilde{\mathrm{q}})}{\tilde{\mathrm{q}}}\mathcal{A}\left({\vartheta }_1^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1},{\mathcal{A}}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\left({\vartheta }_1,{\vartheta }_2\right)\right)\hfill \\ \hfill & -\frac{2{\Delta }_{\tilde{\mathrm{q}}}\left(n,{\tilde{\mathrm{q}}}_{*};{\vartheta }_1,{\vartheta }_2\right)}{\tilde{\mathrm{q}}({\vartheta }_2-{\vartheta }_1)}|\le \frac{({\vartheta }_2-{\vartheta }_1){\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1\right]}_{\tilde{\mathrm{q}}}}{2{(1+\tilde{\mathrm{q}})}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}}\hfill \\ \hfill & \times \left\{{\mathrm{R}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}\left(\frac{1}{3},p\right){\left({\vartheta }_1^{n+1}+{\mathcal{A}}^{n+1}({\vartheta }_1,{\vartheta }_2)\right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}+{\mathrm{R}}_{\tilde{\mathrm{q}}}^{\frac{1}{p}}\left(\frac{2}{3},p\right){\left({\mathcal{A}}^{n+1}({\vartheta }_1,{\vartheta }_2)+{\vartheta }_2^{n+1}\right)}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\right\}.\hfill \end{array}$$

(39)

Proof. 

By applying Corollary 6 for $\eta =\sigma =\frac{1}{2},\frac{1}{3},$ respectively, with $\mathrm{Y}\left(\varsigma \right)=\frac{{\varsigma }^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}}{{\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1\right]}_{\tilde{\mathrm{q}}}}$ for all $\varsigma \in [{\vartheta }_1,{\vartheta }_2]$ and $\alpha =s=1,$ then we can obtain the desired results (38) and (39). □

Proposition 2.

Let $n\in \mathbb{N},\tilde{\mathrm{q}}\in (0,1)$ and ${\vartheta }_1,{\vartheta }_2\in \mathbb{R},$ where $0\le {\vartheta }_1<{\vartheta }_2.$ Then, for ${\tilde{\mathrm{q}}}_{*}\ge 1,$ we have

$$\begin{array}{cc}\hfill & |\mathcal{A}\left({\vartheta }_1^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1},{\vartheta }_2^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\right)+{\mathcal{A}}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\left({\vartheta }_1,{\vartheta }_2\right)+\frac{2(1-\tilde{\mathrm{q}})}{\tilde{\mathrm{q}}}\mathcal{A}\left({\vartheta }_1^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1},{\mathcal{A}}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\left({\vartheta }_1,{\vartheta }_2\right)\right)-\frac{2{\Delta }_{\tilde{\mathrm{q}}}\left(n,{\tilde{\mathrm{q}}}_{*};{\vartheta }_1,{\vartheta }_2\right)}{\tilde{\mathrm{q}}({\vartheta }_2-{\vartheta }_1)}|\hfill \\ \hfill & \le \frac{({\vartheta }_2-{\vartheta }_1){\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1\right]}_{\tilde{\mathrm{q}}}}{2}{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\frac{1}{2}\right)\hfill \\ \hfill & \times \left\{{\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{1}{2}\right){\vartheta }_1^{n+1}+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{1}{2}\right){\mathcal{A}}^{n+1}({\vartheta }_1,{\vartheta }_2)\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}+{\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{1}{2}\right){\mathcal{A}}^{n+1}({\vartheta }_1,{\vartheta }_2)+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{1}{2}\right){\vartheta }_2^{n+1}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\right\},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & |\frac{2}{3}\left[\mathcal{A}\left({\vartheta }_1^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1},{\vartheta }_2^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\right)+2{\mathcal{A}}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\left({\vartheta }_1,{\vartheta }_2\right)\right]+\frac{2(1-\tilde{\mathrm{q}})}{\tilde{\mathrm{q}}}\mathcal{A}\left({\vartheta }_1^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1},{\mathcal{A}}^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}\left({\vartheta }_1,{\vartheta }_2\right)\right)\hfill \\ \hfill & -\frac{2{\Delta }_{\tilde{\mathrm{q}}}\left(n,{\tilde{\mathrm{q}}}_{*};{\vartheta }_1,{\vartheta }_2\right)}{\tilde{\mathrm{q}}({\vartheta }_2-{\vartheta }_1)}|\le \frac{({\vartheta }_2-{\vartheta }_1){\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1\right]}_{\tilde{\mathrm{q}}}}{2}\hfill \\ \hfill & \times \{{\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\frac{1}{3}\right){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{1}{3}\right){\vartheta }_1^{n+1}+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{1}{3}\right){\mathcal{A}}^{n+1}({\vartheta }_1,{\vartheta }_2)\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\hfill \\ \hfill & + {\mathrm{M}}_{\tilde{\mathrm{q}}}^{1-\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\left(\frac{2}{3}\right){\left[{\mathrm{C}}_{\tilde{\mathrm{q}}}\left(\frac{2}{3}\right){\mathcal{A}}^{n+1}({\vartheta }_1,{\vartheta }_2)+{\mathrm{E}}_{\tilde{\mathrm{q}}}\left(\frac{2}{3}\right){\vartheta }_2^{n+1}\right]}^{\frac{1}{{\tilde{\mathrm{q}}}_{*}}}\}.\hfill \end{array}$$

(41)

Proof. 

By using Corollary 13 for $\eta =\sigma =\frac{1}{2},\frac{1}{3},$ respectively, with $\mathrm{Y}\left(\varsigma \right)=\frac{{\varsigma }^{\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1}}{{\left[\frac{n+1}{{\tilde{\mathrm{q}}}_{*}}+1\right]}_{\tilde{\mathrm{q}}}}$ for all $\varsigma \in [{\vartheta }_1,{\vartheta }_2]$ and $\alpha =s=1,$ then we can obtain the desired results (40) and (41). □

6. Conclusions

Convexity performs an essential role in the theoretical field of inequalities. In this article, we established a new parameterized quantum fractional integral identity. By using this, we have explored some quantum fractional integral inequalities of trapezium, midpoint and Simpson’s type via s-convex functions. The Hölders inequality is utilized to obtain the main findings, which has strong applicability in theory of inequalities. Some special cases of our main results are discussed in detail. In order to illustrate the efficiency of the main results, an example and application to special means of positive real numbers are also provided. Numerical analysis investigation shows that the mixed FC with QC yields better estimates compared with FC or QC separately. Interested readers can use $\tilde{\mathrm{q}}$-deformed real numbers [49] to extend our results. We believe that this novel idea that mixed together FC and QC opens many avenues for interested researchers working in these fields and they can discovering further approximations for different kinds of convexity.