New Variants of Quantum Midpoint-Type Inequalities

Abstract

Recently, there has been a strong push toward creating and expanding quadrature inequalities in quantum calculus. In order to investigate various avenues for quantum inquiry, a number of quantum extensions of midpoint estimations are studied. The goal of this research article is to discover novel quantum midpoint-type inequalities that are twice ${\mathfrak{q}}^{{\xi }_2}$-differentiable for $(\alpha ,m)$-convex functions. Firstly, we obtain novel identity for ${\mathfrak{q}}^{{\xi }_2}$-integral by employing quantum calculus tools. Then by using the auxiliary identity, we formulate new bounds by taking into account the known quantum Hölder and Power mean inequalities. An example is provided with a graphical representation to show the validity of obtaining results. The outcomes of this study clarify and expand earlier research on midpoint-type inequalities. Analytic inequalities of this type as well as particularly related strategies have applications for various fields where symmetry plays an important role.

1. Introduction

Calculus without limits, or quantum calculus, is an infinitesimal calculus that has a number of applications. Due to the vast number of applications in areas of mathematics such as fundamental hypergeometric functions, number theory, combinatorics, orthogonal polynomials, physics, relativity, and quantum science, $\mathfrak{q}$-analysis has recently been the topic of a lot of research [1,2,3,4]. It is believed that Euler invented this branch of mathematics by utilizing the $\mathfrak{q}$-parameter in Newton’s infinitive series work. Jackson was the one who first introduced the $\mathfrak{q}$-calculus [1]. Jackson [3] introduced $\mathfrak{q}$-definite integrals in the nineteenth century and started his work in a symmetrical fashion. Agarwal [5] initially presented the $\mathfrak{q}$-fractional derivative in 1969.

The science of differential equations heavily relies on integral inequalities. Numerous academics have investigated applications of integral inequality in both conventional and quantum calculus. Tariboon et al. in [6] uses ${}_{{\xi }_1}{D}_{\mathfrak{q}}$-difference operator and provided applications to impulsive difference equations on finite quantum intervals. In light of the historical knowledge of the importance of mathematical inequalities. Tariboon et al. in [7] introduced the quantum variants of well know integral inequalities such as Hermite-Hadamard, Jensen, Ostrowski, Cebysev and Grüss type inequalities as they performed an essential role in quantum calculus.

The connection between inequalities and convex functions has been discovered to be extremely useful in developing new integral inequalities along with their significant applications [8,9,10]. A convex function plays a notable role in both theoretical and applied sciences. Convexity also has the finest effect on our daily life through countless implementations in medicine, industry, and business. Due to the wide range of implementations, it is among the most advanced branches of mathematical modelling. In literature, serval kinds of convexities are introduced depending on their useful strengths and general nature. However, our study requires the classical convex functions and $(\alpha ,m)$-convex functions defined as:

Definition 1

([8]). If $\mathfrak{G}:[{\xi }_1,{\xi }_2]\subseteq \mathbb{R}\to \mathbb{R}$ is convex, then the inequality

$$\mathfrak{G}(\lambda \mathfrak{x}+(1-\lambda \left)\mathrm{y}\right)\le \lambda \mathfrak{G}\left(\mathfrak{x}\right)+(1-\lambda )\mathfrak{G}\left(\mathrm{y}\right)$$

is valid for all $\mathfrak{x},\mathrm{y}\in [{\xi }_1,{\xi }_2]$ and $\lambda \in [0,1].$

Definition 2

([11]). If $\mathfrak{G}:[0,{\xi }_2)\to \mathbb{R}$ is called $(\alpha ,m)$-convex, then the inequality

$$\mathfrak{G}(\lambda \mathfrak{x}+m(1-\lambda )\mathrm{y})\le {\lambda }^{\alpha }\mathfrak{G}\left(\mathfrak{x}\right)+m(1-{\lambda }^{\alpha })\mathfrak{G}\left(\mathrm{y}\right)$$

holds for all $\mathfrak{x},\mathrm{y}\in [0,{\xi }_2),\lambda \in [0,1],(\alpha ,m)\in (0,1].$

In order to move further we required some basic notions and background of quantum calculus.

2. Description of Quantum Calculus

Some basic known definitions of $\mathfrak{q}$-calculus are presented below:

Definition 3

([4,7]). If $\mathfrak{G}:[{\xi }_1,{\xi }_2]\to mathbbR,$ the ${\mathfrak{q}}_{{\xi }_1}$-derivative of $\mathfrak{G}$ at $\in [{\xi }_1,{\xi }_2]$ is defined as:

$${}_{{\xi }_1}{D}_{\mathfrak{q}}\mathfrak{G}\left(\mathfrak{x}\right)=\frac{\mathfrak{G}\left(\mathfrak{x}\right)-\mathfrak{G}(\mathfrak{q}\mathfrak{x}+(1-\mathfrak{q}){\xi }_1)}{(1-\mathfrak{q})(\mathfrak{x}-{\xi }_1)},\mathfrak{x}\ne {\xi }_1.$$

(1)

If $\mathfrak{x}={\xi }_1$, we define ${\displaystyle {}_{{\xi }_1}{D}_{\mathfrak{q}}\varphi \left({\xi }_1\right)=\underset{\mathfrak{x}\to {\xi }_1}{lim}{}_{{\xi }_1}{D}_{\mathfrak{q}}\varphi \left(\mathfrak{x}\right)}$ if it exists.

Rajkovic [12] defined the Riemann $\mathfrak{q}$-integral which was then generalised to Jackson $\mathfrak{q}$-integral on $[{\xi }_1,{\xi }_2]$:

$$\underset{{\xi }_{\mathfrak{1}}}{\overset{\mathfrak{x}}{\int }}\mathfrak{G}\left(\lambda \right){}_{{\xi }_{\mathfrak{1}}}{d}_{\mathfrak{q}}\lambda =(1-\mathfrak{q})(\mathfrak{x}-{\xi }_1)\sum _{\mathfrak{n}=0}^{\infty }{\mathfrak{q}}^{\mathfrak{n}}\mathfrak{G}({\mathfrak{q}}^{\mathfrak{n}}\mathfrak{x}+(1-{\mathfrak{q}}^{\mathfrak{n}}){\xi }_1),\mathfrak{x}\in [{\xi }_1,{\xi }_2].$$

(2)

Definition 4.

If${\xi }_1=0$in (2), then

$$\underset{0}{\overset{\mathfrak{x}}{\int }}\mathfrak{G}\left(\lambda \right){}_0{d}_{\mathfrak{q}}\lambda =\underset{0}{\overset{\mathfrak{x}}{\int }}\mathfrak{G}\left(\lambda \right)d_{\mathfrak{q}}\lambda ,$$

where $\underset{0}{\overset{\mathfrak{x}}{\int }}\mathfrak{G}\left(\lambda \right)d_{\mathfrak{q}}\lambda$ is $\mathfrak{q}$-definite integral on $[0,\mathfrak{x}]$ and defined as [4]:

$$\underset{0}{\overset{\mathfrak{x}}{\int }}\mathfrak{G}\left(\lambda \right){}_0{d}_{\mathfrak{q}}\lambda =\underset{0}{\overset{\mathfrak{x}}{\int }}\mathfrak{G}\left(\lambda \right)d_{\mathfrak{q}}\lambda =(1-\mathfrak{q})\mathfrak{x}\sum _{\mathfrak{n}=0}^{\infty }{\mathfrak{q}}^{\mathfrak{n}}\mathfrak{G}\left({\mathfrak{q}}^{\mathfrak{n}}\mathfrak{x}\right).$$

(3)

If $c\in \left({\xi }_1,\mathfrak{x}\right)$, then the $\mathfrak{q}$-definite integral on $[c,\mathfrak{x}]$ is expressed as:

$$\underset{c}{\overset{\mathfrak{x}}{\int }}\mathfrak{G}\left(\lambda \right){}_{{\xi }_1}{d}_{\mathfrak{q}}\lambda =\underset{{\xi }_1}{\overset{\mathfrak{x}}{\int }}\mathfrak{G}\left(\lambda \right){}_{{\xi }_1}{d}_{\mathfrak{q}}\lambda -\underset{{\xi }_1}{\overset{c}{\int }}\mathfrak{G}\left(\lambda \right){}_{{\xi }_1}{d}_{\mathfrak{q}}\lambda .$$

(4)

In [13], $\mathfrak{q}$-Hermite–Hadamard inequality was established by Alp et al. and stated as:

Theorem 1.

Let $\mathfrak{G}:[{\xi }_1,{\xi }_2]\to \mathbb{R}$ be a convex function on $[{\xi }_1,{\xi }_2]$ and $\mathfrak{q}\in (0,1)$, we have

$$\mathfrak{G}\left(\frac{\mathfrak{q}{\xi }_1+{\xi }_2}{1+\mathfrak{q}}\right)\le \frac{1}{{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{{\xi }_2}{\int }}\mathfrak{G}\left(\mathfrak{x}\right){}_{{\xi }_1}{d}_{\mathfrak{q}}\mathfrak{x}\le \frac{\mathfrak{q}\mathfrak{G}\left({\xi }_1\right)+\mathfrak{G}\left({\xi }_2\right)}{1+\mathfrak{q}}.$$

In [14], authors presented some new definitions of quantum calculus and also presented a novel variant of Hermite-Hadamard inequality:

Definition 5

([14]). If $\mathfrak{G}:[{\xi }_1,{\xi }_2]\to \mathbb{R}$ be a function, then ${\mathfrak{q}}^{{\xi }_2}$-definite integral on $[{\xi }_1,{\xi }_2]$ is expressed as:

$$\begin{array}{cc}\hfill \underset{{\xi }_1}{\overset{{\xi }_2}{\int }}\mathfrak{G}\left(\mathfrak{x}\right){}^{{\xi }_2}{d}_{\mathfrak{q}}\mathfrak{x}& =(1-\mathfrak{q})({\xi }_2-{\xi }_1)\sum _{\mathfrak{n}=0}^{\infty }{\mathfrak{q}}^{\mathfrak{n}}\mathfrak{G}({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+(1-{\mathfrak{q}}^{\mathfrak{n}}){\xi }_2)\hfill \\ \hfill & =({\xi }_2-{\xi }_1)\underset{0}{\overset{1}{\int }}\mathfrak{G}(\lambda {\xi }_1+(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda .\hfill \end{array}$$

(5)

Definition 6

([14]). If $\mathfrak{G}:[{\xi }_1,{\xi }_2]\to \mathbb{R}$ be a function, then ${\mathfrak{q}}^{{\xi }_2}$-derivative of $\mathfrak{G}$ at $\mathfrak{x}\in [{\xi }_1,{\xi }_2]$ is expressed as:

$${}^{{\xi }_2}{D}_{\mathfrak{q}}\mathfrak{G}\left(\mathfrak{x}\right)=\frac{\mathfrak{G}(\mathfrak{q}\mathfrak{x}+(1-\mathfrak{q}){\xi }_2)-\mathfrak{G}\left(\mathfrak{x}\right)}{(1-\mathfrak{q})({\xi }_2-\mathfrak{x})},\mathfrak{x}\ne {\xi }_2.$$

Theorem 2

([14]). Let $\mathfrak{G}:[{\xi }_1,{\xi }_2]\to \mathbb{R}$ is a convex function on $[{\xi }_1,{\xi }_2]$ and $\mathfrak{q}\in (0,1)$, then $\mathfrak{q}$-Hermite–Hadamard inequalities are given bellow:

$$\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}{\xi }_2}{1+\mathfrak{q}}\right)\le \frac{1}{{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{{\xi }_2}{\int }}\mathfrak{G}\left(\mathfrak{x}\right){}^{{\xi }_2}{d}_{\mathfrak{q}}\mathfrak{x}\le \frac{\mathfrak{G}\left({\xi }_1\right)+\mathfrak{q}\mathfrak{G}\left({\xi }_2\right)}{1+\mathfrak{q}}.$$

(6)

The following notations were frequently used:

$${\left[\mathfrak{n}\right]}_{\mathfrak{q}}=\sum _{i=0}^{\mathfrak{n}-1}{\mathfrak{q}}^i$$

and

$${(1-\lambda )}_{\mathfrak{q}}^{\mathfrak{n}}={(\lambda ,\mathfrak{q})}_{\mathfrak{n}}=\prod _{i=0}^{\mathfrak{n}-1}(1-{\mathfrak{q}}^i\lambda ),$$

(7)

where $\mathfrak{q}\in (0,1).$

Lemma 1

([13]). Following equality holds:

$$\underset{{\xi }_1}{\overset{\mathfrak{x}}{\int }}{(\lambda -{\xi }_1)}^{\alpha }{}_{{\xi }_1}{d}_{\mathfrak{q}}\lambda =\frac{{(\mathfrak{x}-{\xi }_1)}^{\alpha +1}}{[\alpha +1]{}_{\mathfrak{q}}},$$

(8)

for $\alpha \in \mathbb{R}\backslash \{-1\}$.

Lemma 2

([15]). Following equality holds:

$$\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^{\mathfrak{n}}{d}_{\mathfrak{q}}\lambda =\frac{{(1-\frac{1}{{\left[2\right]}_{\mathfrak{q}}})}_{\mathfrak{q}}^{\mathfrak{n}+1}}{{[\mathfrak{n}+1]}_{\mathfrak{q}}}.$$

The $\mathfrak{q}$-analogues of integral inequalities are a subject of utmost interest while concentrating on the exhilaration and enchantment of the emergence of $\mathfrak{q}$-calculus and its applicability in mathematical physics. Taking into account ${}_{{\xi }_1}{D}_{\mathfrak{q}}$-derivative, ${\mathfrak{q}}_{{\xi }_1}$-integral, ${}^{{\xi }_2}{D}_{\mathfrak{q}}$-derivative and ${\mathfrak{q}}^{{\xi }_2}$-integral, quantum variants of notable integral inequalities have been investigated pertaining various kinds of convexity (see [7,16,17,18,19]). Some quantum inequalities for coordinate convex functions can be seen in [20,21]. However, the notable bounds via $\mathfrak{q}$-midpoint inequalities can be observed in [13,15,22].

The compelling goal of this study, which was inspired by the current trend, is to set new limits for $\mathfrak{q}$-midpoint inequalities. Firstly, we established an auxiliary identity pertaining to twice different quantum functions. Then, by employing $(\alpha ,m)$-convexity to this identity, we gain some new results for quantum midpoint inequalities for twice $\mathfrak{q}$-differentiable functions. By choosing $\alpha =1$, $m=1$ and taking $\mathfrak{q}\to 1^{-}$, we can also recapture the findings in a classical sense.

3. New Quantum Midpoint Type Identity for Twice $\mathfrak{q}$-Differentiable Functions

We will demonstrate equality in this section which help us in achieving our main goals.

Lemma 3.

Let $\mathfrak{G}:[{\xi }_1,{\xi }_2]\to \Re$ be a $\mathfrak{q}$-differentiable function on $({\xi }_1,{\xi }_2)$. If ${}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}$ is continuous and integrable on $[{\xi }_1,{\xi }_2]$, we attain the identity:

$$\begin{array}{cc}\hfill & \frac{{(m{\xi }_2-{\xi }_1)}^2}{{\left[2\right]}_{\mathfrak{q}}}\left[\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{\mathfrak{q}}^3{\lambda }^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda \right.\hfill \\ \hfill & \left.+\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda \right]\hfill \\ \hfill & =\frac{1}{(m{\xi }_2-{\xi }_1)}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right){}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right).\hfill \end{array}$$

Proof. 

By using Definition 6, we attain the following equality

$$\begin{array}{cc}\hfill & {}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)\hfill \\ \hfill & ={}^{{\xi }_2}{D}_{\mathfrak{q}}\left({}^{{\xi }_2}{D}_{\mathfrak{q}}\left(\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)\right)\right)\hfill \\ \hfill & ={}^{{\xi }_2}{D}_{\mathfrak{q}}\left(\frac{\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)-\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)}{(1-\mathfrak{q})({\xi }_2-{\xi }_1)\lambda }\right)\hfill \\ \hfill & =\frac{1}{(1-\mathfrak{q})({\xi }_2-{\xi }_1)\lambda }\left[\frac{\mathfrak{G}({\mathfrak{q}}^2\lambda {\xi }_1+m(1-\lambda {\mathfrak{q}}^2){\xi }_2)-\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)}{(1-\mathfrak{q})\mathfrak{q}({\xi }_2-{\xi }_1)\lambda }\right.\hfill \\ \hfill & \left.-\frac{\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)-\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)}{(1-\mathfrak{q})({\xi }_2-{\xi }_1)\lambda }\right]\hfill \\ \hfill & =\frac{\mathfrak{G}({\mathfrak{q}}^2\lambda {\xi }_1+m(1-\lambda {\mathfrak{q}}^2){\xi }_2)-\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)}{{(1-\mathfrak{q})}^2\mathfrak{q}{({\xi }_2-{\xi }_1)}^2{\lambda }^2}\hfill \\ \hfill & -\frac{\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)-\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)}{{(1-\mathfrak{q})}^2{({\xi }_2-{\xi }_1)}^2{\lambda }^2}\hfill \\ \hfill & =\frac{\mathfrak{G}({\mathfrak{q}}^2\lambda {\xi }_1+m(1-\lambda {\mathfrak{q}}^2){\xi }_2)-(1+\mathfrak{q})\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)+\mathfrak{q}\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda )m{\xi }_2)}{{(1-\mathfrak{q})}^2\mathfrak{q}{({\xi }_2-{\xi }_1)}^2{\lambda }^2}.\hfill \end{array}$$

(9)

From (9) and using the fundamental properties of $\mathfrak{q}$-integral, we have

$$\begin{array}{cc}\hfill & {\int }_0^{1/{\left[2\right]}_{\mathfrak{q}}}{\mathfrak{q}}^3{\lambda }^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda +\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & =\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{\mathfrak{q}}^3{\lambda }^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda +\underset{0}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & -\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & =\underset{0}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & +\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}({\mathfrak{q}}^3{\lambda }^2-{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2){}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & =\frac{1}{{(1-\mathfrak{q})}^2{(\mathfrak{m}{\xi }_{\mathfrak{2}}-{\xi }_1)}^2}\underset{0}{\overset{1}{\int }}\frac{{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2}{{\lambda }^2}\left(\frac{1}{\mathfrak{q}}\mathfrak{G}\left(({\mathfrak{q}}^2\lambda {\xi }_1+m(1-{\mathfrak{q}}^2\lambda ){\xi }_2)\right)\right.\hfill \\ \hfill & \left.-\frac{1+\mathfrak{q}}{\mathfrak{q}}\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)+\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)\right)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & +\frac{1}{{(1-\mathfrak{q})}^2{(m{\xi }_2-{\xi }_1)}^2}{\int }_0^{1/{\left[2\right]}_{\mathfrak{q}}}\frac{({\mathfrak{q}}^3{\lambda }^2-{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2)}{{\lambda }^2}\left(\frac{1}{\mathfrak{q}}\mathfrak{G}\left(({\mathfrak{q}}^2\lambda {\xi }_1+m(1-{\mathfrak{q}}^2\lambda ){\xi }_2)\right)\right.\hfill \\ \hfill & \left.-\frac{1+\mathfrak{q}}{\mathfrak{q}}\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)+\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)\right)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & =\frac{[I_1+I_2]}{{(1-\mathfrak{q})}^2{(m{\xi }_2-{\xi }_1)}^2}.\hfill \end{array}$$

Calculate the values of integrals $I_1$ and $I_2$ in the following way:

$$\begin{array}{cc}\hfill I_1& =\underset{0}{\overset{1}{\int }}\frac{{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2}{{\lambda }^2}\left(\frac{1}{\mathfrak{q}}\mathfrak{G}\left({\mathfrak{q}}^2\lambda {\xi }_1+m(1-{\mathfrak{q}}^2\lambda ){\xi }_2\right)-\frac{1+\mathfrak{q}}{\mathfrak{q}}\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)\right.\hfill \\ \hfill & \left.+\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)\right)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & =(1-\mathfrak{q})\sum _{\mathfrak{n}=0}^{\infty }{\mathfrak{q}}^{\mathfrak{n}}\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{2\mathfrak{n}}}\left\{\frac{1}{\mathfrak{q}}\mathfrak{G}\left({\mathfrak{q}}^2{\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m(1-{\mathfrak{q}}^2{\mathfrak{q}}^{\mathfrak{n}}){\xi }_2\right)\right.\hfill \\ \hfill & \left.-\frac{1+\mathfrak{q}}{\mathfrak{q}}\mathfrak{G}(\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}}){\xi }_2)+\mathfrak{G}({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m(1-{\mathfrak{q}}^{\mathfrak{n}}){\xi }_2)\right\}\hfill \\ \hfill & =(1-\mathfrak{q})\left\{\frac{1}{\mathfrak{q}}\sum _{\mathfrak{n}=0}^{\infty }\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}}}\mathfrak{G}\left(({\mathfrak{q}}^{\mathfrak{n}+2}{\xi }_1+mb(1-{\mathfrak{q}}^{\mathfrak{n}+2}){\xi }_2)\right)\right.\hfill \\ \hfill & -\frac{1+\mathfrak{q}}{\mathfrak{q}}\sum _{\mathfrak{n}=0}^{\infty }\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}}}\mathfrak{G}({\mathfrak{q}}^{\mathfrak{n}+1}{\xi }_1+m(1-{\mathfrak{q}}^{\mathfrak{n}+1}){\xi }_2)\hfill \\ \hfill & \left.+\sum _{\mathfrak{n}=0}^{\infty }\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}}}\mathfrak{G}({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m(1-{\mathfrak{q}}^{\mathfrak{n}}){\xi }_2)\right\}\hfill \\ \hfill & =\frac{(1-\mathfrak{q})}{\mathfrak{q}}\sum _{\mathfrak{n}=2}^{\infty }\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}-2})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}-2}}\mathfrak{G}\left(({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m(1-{\mathfrak{q}}^{\mathfrak{n}}){\xi }_2)\right)\hfill \\ \hfill & -\frac{(1-\mathfrak{q})(1+\mathfrak{q})}{\mathfrak{q}}\sum _{\mathfrak{n}=1}^{\infty }\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}-1})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}-1}}\mathfrak{G}({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m(1-{\mathfrak{q}}^{\mathfrak{n}}){\xi }_2)\hfill \\ \hfill & +(1-\mathfrak{q})\sum _{\mathfrak{n}=0}^{\infty }\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}}}\mathfrak{G}({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m(1-{\mathfrak{q}}^{\mathfrak{n}}){\xi }_2)\hfill \\ \hfill & =\frac{\left(1-\mathfrak{q}\right)}{\mathfrak{q}}\sum _{\mathfrak{n}=0}^{\infty }\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}-2})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}-2}}\mathfrak{G}\left(\left({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m\left(1-{\mathfrak{q}}^{\mathfrak{n}}\right){\xi }_2\right)\right)\hfill \\ \hfill & -\frac{\left(1-\mathfrak{q}\right)\left(1+\mathfrak{q}\right)}{\mathfrak{q}}\sum _{\mathfrak{n}=0}^{\infty }\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}-1})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}-1}}\mathfrak{G}\left({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m\left(1-{\mathfrak{q}}^{\mathfrak{n}}\right){\xi }_2\right)\hfill \\ \hfill & +\left(1-\mathfrak{q}\right)\sum _{\mathfrak{n}=0}^{\infty }\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}}}\mathfrak{G}\left({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m\left(1-{\mathfrak{q}}^{\mathfrak{n}}\right){\xi }_2\right)+\frac{\left(1-\mathfrak{q}\right)\left(1+\mathfrak{q}\right)}{\mathfrak{q}}\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{-1})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{-1}}\mathfrak{G}\left({\xi }_1\right)\hfill \\ \hfill & -\frac{\left(1-\mathfrak{q}\right)}{\mathfrak{q}}\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{-2})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{-2}}\mathfrak{G}\left({\xi }_1\right)-\frac{\left(1-\mathfrak{q}\right)}{\mathfrak{q}}\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{-1})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{-1}}\mathfrak{G}\left(\left(\mathfrak{q}{\xi }_1+m\left(1-\mathfrak{q}\right){\xi }_2\right)\right)\hfill \\ \hfill & =\left(1-\mathfrak{q}\right)\sum _{\mathfrak{n}=0}^{\infty }\left\{\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}-2})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}-1}}-\frac{\left(1+\mathfrak{q}\right)}{\mathfrak{q}}\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}-1})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}-1}}+\frac{{(1-\mathfrak{q}{\mathfrak{q}}^{\mathfrak{n}})}_{\mathfrak{q}}^2}{{\mathfrak{q}}^{\mathfrak{n}}}\right\}\hfill \\ \hfill & \times \mathfrak{G}\left({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m\left(1-{\mathfrak{q}}^{\mathfrak{n}}\right){\xi }_2\right)\hfill \\ \hfill & +\left[\left(1-{\mathfrak{q}}^2\right){(1-\mathfrak{q}{\mathfrak{q}}^{-1})}_{\mathfrak{q}}^2-\mathfrak{q}\left(1-\mathfrak{q}\right){(1-\mathfrak{q}{\mathfrak{q}}^{-2})}_{\mathfrak{q}}^2\right]\mathfrak{G}\left({\xi }_1\right)\hfill \\ \hfill & -\left(1-\mathfrak{q}\right){(1-\mathfrak{q}{\mathfrak{q}}^{-1})}_{\mathfrak{q}}^2\mathfrak{G}\left(\mathfrak{q}{\xi }_1+m\left(1-\mathfrak{q}\right){\xi }_2\right)\hfill \\ \hfill & =\left(1+\mathfrak{q}\right){\left(1-\mathfrak{q}\right)}^2\left(1-\mathfrak{q}\right)\sum _{\mathfrak{n}=0}^{\infty }{\mathfrak{q}}^{\mathfrak{n}}\mathfrak{G}\left({\mathfrak{q}}^{\mathfrak{n}}{\xi }_1+m\left(1-{\mathfrak{q}}^{\mathfrak{n}}\right){\xi }_2\right)\hfill \\ \hfill & =\frac{\left(1+\mathfrak{q}\right){\left(1-\mathfrak{q}\right)}^2}{(m{\xi }_2-{\xi }_1)}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right){}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda .\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill I_2& =\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}\frac{({\mathfrak{q}}^3{\lambda }^2-{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2)}{{\lambda }^2}\left(\frac{1}{\mathfrak{q}}\mathfrak{G}\left(({\mathfrak{q}}^2\lambda {\xi }_1+m(1-{\mathfrak{q}}^2\lambda ){\xi }_2)\right)\right.\hfill \\ \hfill & \left.-\frac{1+\mathfrak{q}}{\mathfrak{q}}\mathfrak{G}(\mathfrak{q}\lambda {\xi }_1+m(1-\mathfrak{q}\lambda ){\xi }_2)+\mathfrak{G}(\lambda {\xi }_1+m(1-\lambda ){\xi }_2)\right)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & =\frac{(1-\mathfrak{q})}{{\left[2\right]}_{\mathfrak{q}}}\sum _{\mathfrak{n}=0}^{\infty }\frac{\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)}{{\mathfrak{q}}^{\mathfrak{n}}}\hfill \\ \hfill & \times \left(\frac{1}{\mathfrak{q}}\mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}+2}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}+2}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)-\frac{1+\mathfrak{q}}{\mathfrak{q}}\mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}+1}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}+1}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)\right.\hfill \\ \hfill & +\left.\mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)\right)\hfill \\ \hfill & =\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\sum _{\mathfrak{n}=0}^{\infty }\frac{\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)}{{\mathfrak{q}}^{\mathfrak{n}+1}}\mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}+2}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}+2}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)\hfill \\ \hfill & -\frac{\left(1-\mathfrak{q}\right)\left(1+\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\sum _{\mathfrak{n}=0}^{\infty }\frac{\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)}{{\mathfrak{q}}^{\mathfrak{n}+1}}\mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}+1}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}+1}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)\hfill \\ \hfill & +\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\sum _{\mathfrak{n}=0}^{\infty }\frac{\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)}{{\mathfrak{q}}^{\mathfrak{n}}}\mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)\hfill \\ \hfill & =\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\sum _{\mathfrak{n}=0}^{\infty }\frac{\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}-4}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}-2}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)}{{\mathfrak{q}}^{\mathfrak{n}-1}}\mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)\hfill \\ \hfill & -\frac{\left(1-\mathfrak{q}\right)\left(1+\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\sum _{\mathfrak{n}=0}^{\infty }\frac{\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}-2}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}-1}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)}{{\mathfrak{q}}^{\mathfrak{n}}}\mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)\hfill \\ \hfill & +\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\sum _{\mathfrak{n}=0}^{\infty }\frac{\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)}{{\mathfrak{q}}^{\mathfrak{n}}}\mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)\hfill \\ \hfill & -\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\frac{\left({\mathfrak{q}}^3{\mathfrak{q}}^{-4}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{-2}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)}{{\mathfrak{q}}^{-1}}\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\hfill \\ \hfill & -\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\left({\mathfrak{q}}^3{\mathfrak{q}}^{-2}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{-1}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)\mathfrak{G}\left(\frac{\mathfrak{q}{\xi }_1+m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\hfill \\ \hfill & +\frac{\left(1-\mathfrak{q}\right)\left(1+\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\left({\mathfrak{q}}^3{\mathfrak{q}}^{-2}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{-1}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\hfill \\ \hfill & =\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\sum _{\mathfrak{n}=0}^{\infty }\frac{1}{{\mathfrak{q}}^{\mathfrak{n}}}\left\{\begin{array}{c}\mathfrak{q}\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}-4}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}-2}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)\\ -\left(1+\mathfrak{q}\right)\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}-2}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}-1}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)\\ +\left({\mathfrak{q}}^3{\mathfrak{q}}^{2\mathfrak{n}}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)\end{array}\right\}\hfill \\ \hfill & \times \mathfrak{G}\left(\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}{\xi }_1+m\left(1-\frac{{\mathfrak{q}}^{\mathfrak{n}}}{{\left[2\right]}_{\mathfrak{q}}}\right){\xi }_2\right)\hfill \\ \hfill & +\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\left\{\begin{array}{c}\left(1+\mathfrak{q}\right)\left({\mathfrak{q}}^3{\mathfrak{q}}^{-2}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{-1}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)\\ -\mathfrak{q}\left({\mathfrak{q}}^3{\mathfrak{q}}^{-4}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{-2}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)\end{array}\right\}\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\hfill \\ \hfill & -\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\left({\mathfrak{q}}^3{\mathfrak{q}}^{-2}-{\left[2\right]}_{\mathfrak{q}}^2{\left(1-\mathfrak{q}\frac{{\mathfrak{q}}^{-1}}{{\left[2\right]}_{\mathfrak{q}}}\right)}_{\mathfrak{q}}^2\right)\mathfrak{G}\left(\frac{\mathfrak{q}{\xi }_1+m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\hfill \\ \hfill & =\frac{\left(1-\mathfrak{q}\right)}{{\left[2\right]}_{\mathfrak{q}}}\left\{\left[-1+{\mathfrak{q}}^2+{\mathfrak{q}}^3-\mathfrak{q}\right]\right\}\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\hfill \\ \hfill & =-{\left(1-\mathfrak{q}\right)}^2\left(1+\mathfrak{q}\right)\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right).\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill & \underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{\mathfrak{q}}^3{\lambda }^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\lambda {\xi }_1+m\left(1-\lambda \right){\xi }_2\right)d_{\mathfrak{q}}\lambda +\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\lambda {\xi }_1+m\left(1-\lambda \right){\xi }_2\right)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & =\frac{1}{{(1-\mathfrak{q})}^2{(m{\xi }_2-{\xi }_1)}^2}\left(\frac{\left(1+\mathfrak{q}\right){\left(1-\mathfrak{q}\right)}^2}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}{\left(\lambda \right)}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -{\left(1-\mathfrak{q}\right)}^2\left(1+\mathfrak{q}\right)\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right)\hfill \\ \hfill & =\frac{{\left[2\right]}_{\mathfrak{q}}}{{\left(m{\xi }_2-{\xi }_1\right)}^2}\left(\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right){}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right),\hfill \end{array}$$

which completes the proof. □

Remark 1.

By setting $\mathfrak{q}\to 1^{-}$ and $m=1$, then we have

$$\begin{array}{cc}\hfill & \frac{{\left({\xi }_2-{\xi }_1\right)}^2}{2}\left[\underset{0}{\overset{1/2}{\int }}{\lambda }^2{\mathfrak{G}}^{″}\left(\lambda {\xi }_1+\left(1-\lambda \right){\xi }_2\right)d\lambda +\underset{1/2}{\overset{1}{\int }}{(1-\lambda )}^2{\mathfrak{G}}^{″}\left(\lambda {\xi }_1+\left(1-\lambda \right){\xi }_2\right)d\lambda \right]\hfill \\ \hfill & =\frac{1}{({\xi }_2-{\xi }_1)}\underset{{\xi }_1}{\overset{{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right)d\lambda -\mathfrak{G}\left(\frac{{\xi }_1+{\xi }_2}{2}\right),\hfill \end{array}$$

which was given in [23].

Remark 2.

By choosing $m=1$, we recapture ([22], Lemma 5).

Quantum Midpoint Type Inequalities

Theorem 3.

Let the assumptions of Lemma 3 hold. If $\left|{}^{{\xi }_{\mathfrak{2}}}{D}_{\mathfrak{q}}^2\mathfrak{G}\right|$ is $(\alpha ,m)$-convex on $[{\xi }_1,{\xi }_2]$, we have the inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right){}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right|\hfill \\ \hfill & \le \frac{{\mathfrak{q}}^3{(m{\xi }_2-{\xi }_1)}^2}{{\left[2\right]}_{\mathfrak{q}}}\left(\frac{{\left[3\right]}_{\mathfrak{q}}\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[3\right]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}+m\frac{({\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +3]}_{\mathfrak{q}}-{\left[3\right]}_{\mathfrak{q}})\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[3\right]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right)\hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}\left(\left[\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +2]}_{\mathfrak{q}}-\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha +2}{[\alpha +1]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right.\right.\hfill \\ \hfill & \left.+\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left(\mathfrak{q}{[\alpha +1]}_{\mathfrak{q}}-{[\alpha +2]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}+\frac{{\mathfrak{q}}^3{[\alpha +2]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right]\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|\hfill \\ \hfill & +m\left[\frac{\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}+1}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +1]}_{\mathfrak{q}}}-\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}\left(\frac{(2\mathfrak{q}+{\mathfrak{q}}^2){\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +2]}_{\mathfrak{q}}-({\left[2\right]}_{\mathfrak{q}}^{\alpha +2}-1){\left[2\right]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +2]}_{\mathfrak{q}}}\right)\right.\hfill \\ \hfill & \left.\left.-{\mathfrak{q}}^3\frac{({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1){\left[3\right]}_{\mathfrak{q}}-(3\mathfrak{q}+3{\mathfrak{q}}^2+{\mathfrak{q}}^3){\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +3]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[3\right]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right]\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|\right).\hfill \end{array}$$

(10)

Proof. 

Taking modulus on Lemma 3, we get

$$\begin{array}{cc}\hfill & \left|\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}{\left(\lambda \right)}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right|\hfill \\ \hfill & \le \frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}\left[\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{\mathfrak{q}}^3{\lambda }^2\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\lambda {\xi }_1+m\left(1-\lambda \right){\xi }_2\right)\right|d_{\mathfrak{q}}\lambda \right.\hfill \\ \hfill & \left.+\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\lambda {\xi }_1+m\left(1-\lambda \right){\xi }_2\right)\right|d_{\mathfrak{q}}\lambda \right].\hfill \end{array}$$

By using $(\alpha ,m)$-convexity of $\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\right|$, we can write

$$\begin{array}{cc}\hfill & \left|\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}{\left(\lambda \right)}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{\left[2\right]{}_{\mathfrak{q}}}\right)\right|\hfill \\ \hfill & \le \frac{{\mathfrak{q}}^3{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}\left({\lambda }^{\alpha +2}\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|+m\left({\lambda }^2-{\lambda }^{\alpha +2}\right)\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|\right)d_{\mathfrak{q}}\lambda \hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}\left({\lambda }^{\alpha }{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|+m\left(1-{\lambda }^{\alpha }\right){(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|\right)d_{\mathfrak{q}}\lambda .\hfill \end{array}$$

(11)

We have the facts that

$$\begin{array}{cc}\hfill \underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}\lambda {}^{\alpha +2}{d}_{\mathfrak{q}}\lambda & =\frac{1}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +3\right]}_{\mathfrak{q}}},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}\left({\lambda }^2-{\lambda }^3\right)d_{\mathfrak{q}}\lambda & =\frac{1}{{\left[2\right]}_{\mathfrak{q}}^3{\left[3\right]}_{\mathfrak{q}}}-\frac{1}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +3\right]}_{\mathfrak{q}}},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{\lambda }^{\alpha }{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{d}_{\mathfrak{q}}& =\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +2]}_{\mathfrak{q}}-\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha +2}{[\alpha +1]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\hfill \\ \hfill & +\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left(\mathfrak{q}{[\alpha +1]}_{\mathfrak{q}}-{[\alpha +2]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}+\frac{{\mathfrak{q}}^3{[\alpha +2]}_{\mathfrak{q}}{[\alpha +1]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\hfill \end{array}$$

(14)

and

$$\begin{array}{cc}\hfill \underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}\left(1-{\lambda }^{\alpha }\right){(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{d}_{\mathfrak{q}}\lambda & =\frac{\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}+1}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +1]}_{\mathfrak{q}}}\hfill \\ \hfill & -\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}\left[\frac{(2\mathfrak{q}+{\mathfrak{q}}^2){\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +2]}_{\mathfrak{q}}-({\left[2\right]}_{\mathfrak{q}}^{\alpha +2}-1){\left[2\right]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +2]}_{\mathfrak{q}}}\right]\hfill \\ \hfill & -{\mathfrak{q}}^3\left[\frac{({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1){\left[3\right]}_{\mathfrak{q}}-(3\mathfrak{q}+3{\mathfrak{q}}^2+{\mathfrak{q}}^3){\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +3]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[3\right]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right].\hfill \end{array}$$

(15)

By putting (12)–(15) into (11), we can achieve the required results. □

Remark 3.

By taking limit $\mathfrak{q}\to 1^{-}$ and $\alpha =m=1$ in Theorem 3, we attain

$$\left|\frac{1}{{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right)d\lambda -\mathfrak{G}\left(\frac{{\xi }_1+{\xi }_2}{2}\right)\right|\le \frac{{\left({\xi }_2-{\xi }_1\right)}^2}{48}\left(\left|{\mathfrak{G}}^{″}\left({\xi }_1\right)\right|+\left|{\mathfrak{G}}^{″}\left({\xi }_2\right)\right|\right),$$

which was given in ([23], Theorem 3).

Remark 4.

By choosing $\alpha =1$ and $m=1$, we recapture ([22], Theorem 3).

For more clarity of results obtained, we provide the following example with graphs ensuring the correctness of the bounds obtained.

Example 1.

Let consider the function $\mathfrak{G}:\left[0,1\right]\to \mathbb{R}$ defined by $\mathfrak{G}\left(\xi \right)={\xi }^3$ and let $m=\frac{1}{2}$ and $\alpha =1.$ Under these assumptions, we have

$$\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right){}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda ={\int }_0^{1/2}{\zeta }^3{}^{\frac{1}{2}}{d}_{\mathfrak{q}}\zeta =\frac{1}{16}\left[1-\frac{3}{{\left[2\right]}_{\mathfrak{q}}}+\frac{3}{{\left[3\right]}_{\mathfrak{q}}}-\frac{1}{{\left[4\right]}_{\mathfrak{q}}}\right]$$

and

$${}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\xi \right)=\left({\left[4\right]}_{\mathfrak{q}}+\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}\right)\xi +\left(2+\mathfrak{q}\right)\left(1-{\mathfrak{q}}^2\right).$$

It is clear that $\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\right|$ is convex on $\left[0,1\right]$. So we can apply Theorem 3 to the function defined by $\mathfrak{G}\left(\xi \right)={\xi }^3.$ Thus, the left hand side of the inequality (10) reduces to

$$\begin{array}{c}\hfill \left|\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right){}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right|=\left|\frac{1}{8}\left[1-\frac{3}{{\left[2\right]}_{\mathfrak{q}}}+\frac{3}{{\left[3\right]}_{\mathfrak{q}}}-\frac{1}{{\left[4\right]}_{\mathfrak{q}}}\right]-\frac{q^3}{8{\left[2\right]}_{\mathfrak{q}}^3}\right|.\end{array}$$

On the other hand, since we have the facts that

$$|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)|=\left(2+\mathfrak{q}\right)\left(1-{\mathfrak{q}}^2\right),$$

and

$$|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)|=3{\left[2\right]}_{\mathfrak{q}}.$$

then the right hand side of the inequality (10) reduces to

$$\begin{array}{cc}\hfill & \frac{{\mathfrak{q}}^3{(m{\xi }_2-{\xi }_1)}^2}{{\left[2\right]}_{\mathfrak{q}}}\left(\frac{{\left[3\right]}_{\mathfrak{q}}\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[3\right]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}+m\frac{({\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +3]}_{\mathfrak{q}}-{\left[3\right]}_{\mathfrak{q}})\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[3\right]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right)\hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}\left(\left[\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +2]}_{\mathfrak{q}}-\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha +2}{[\alpha +1]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right.\right.\hfill \\ \hfill & \left.+\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left(\mathfrak{q}{[\alpha +1]}_{\mathfrak{q}}-{[\alpha +2]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}+\frac{{\mathfrak{q}}^3{[\alpha +2]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right]\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|\hfill \\ \hfill & +m\left[\frac{\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}+1}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +1]}_{\mathfrak{q}}}-\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}\left(\frac{(2\mathfrak{q}+{\mathfrak{q}}^2){\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +2]}_{\mathfrak{q}}-({\left[2\right]}_{\mathfrak{q}}^{\alpha +2}-1){\left[2\right]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +2]}_{\mathfrak{q}}}\right)\right.\hfill \\ \hfill & \left.\left.-{\mathfrak{q}}^3\frac{({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1){\left[3\right]}_{\mathfrak{q}}-(3\mathfrak{q}+3{\mathfrak{q}}^2+{\mathfrak{q}}^3){\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +3]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[3\right]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right]\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|\right)\hfill \\ \hfill & =\frac{3{\mathfrak{q}}^3({\left[2\right]}_{\mathfrak{q}}{\left[4\right]}_{\mathfrak{q}}-{\left[3\right]}_{\mathfrak{q}})}{8{\left[2\right]}_{\mathfrak{q}}^4{\left[3\right]}_{\mathfrak{q}}{\left[4\right]}_{\mathfrak{q}}}+\frac{\left(2+q\right)\left(1-q\right){\mathfrak{q}}^3}{4{\left[2\right]}_{\mathfrak{q}}^4{\left[4\right]}_{\mathfrak{q}}}\hfill \\ \hfill & +\frac{\left(2+\mathfrak{q}\right)\left(1-\mathfrak{q}\right)}{4}\left[\frac{{\left[3\right]}_{\mathfrak{q}}-\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^2}{{\left[2\right]}_{\mathfrak{q}}{\left[3\right]}_{\mathfrak{q}}}-\frac{1}{{\left[2\right]}_{\mathfrak{q}}^3{\left[3\right]}_{\mathfrak{q}}}+\frac{{\mathfrak{q}}^3\left({\left[2\right]}_{\mathfrak{q}}^4-1\right)}{{\left[2\right]}_{\mathfrak{q}}^5{\left[4\right]}_{\mathfrak{q}}}\right]\hfill \\ \hfill & +\frac{3}{2}\left[\frac{\left(\mathfrak{q}-1\right){\left[2\right]}_{\mathfrak{q}}^2+1}{{\left[2\right]}_{\mathfrak{q}}^2}-\frac{\mathfrak{q}(2\mathfrak{q}+{\mathfrak{q}}^2){\left[3\right]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^3+1}{{\left[2\right]}_{\mathfrak{q}}{\left[3\right]}_{\mathfrak{q}}}\right.\hfill \\ \hfill & \left.-{\mathfrak{q}}^3\frac{({\left[2\right]}_{\mathfrak{q}}^4-1){\left[3\right]}_{\mathfrak{q}}-(3\mathfrak{q}+3{\mathfrak{q}}^2+{\mathfrak{q}}^3){\left[2\right]}_{\mathfrak{q}}{\left[4\right]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^3{\left[3\right]}_{\mathfrak{q}}{\left[4\right]}_{\mathfrak{q}}}\right].\hfill \end{array}$$

By the inequality (10), we have the inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left[1-\frac{3}{{\left[2\right]}_q}+\frac{3}{{\left[3\right]}_q}-\frac{1}{{\left[4\right]}_q}\right]-\frac{q^3}{8{\left[2\right]}_q^3}\right|\hfill \\ \hfill & \le \frac{3{\mathfrak{q}}^3({\left[2\right]}_{\mathfrak{q}}{\left[4\right]}_{\mathfrak{q}}-{\left[3\right]}_{\mathfrak{q}})}{8{\left[2\right]}_{\mathfrak{q}}^4{\left[3\right]}_{\mathfrak{q}}{\left[4\right]}_{\mathfrak{q}}}+\frac{\left(2+q\right)\left(1-q\right){\mathfrak{q}}^3}{4{\left[2\right]}_{\mathfrak{q}}^4{\left[4\right]}_{\mathfrak{q}}}\hfill \\ \hfill & +\frac{\left(2+\mathfrak{q}\right)\left(1-\mathfrak{q}\right)}{4}\left[\frac{{\left[3\right]}_{\mathfrak{q}}-\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^2}{{\left[2\right]}_{\mathfrak{q}}{\left[3\right]}_{\mathfrak{q}}}-\frac{1}{{\left[2\right]}_{\mathfrak{q}}^3{\left[3\right]}_{\mathfrak{q}}}+\frac{{\mathfrak{q}}^3\left({\left[2\right]}_{\mathfrak{q}}^4-1\right)}{{\left[2\right]}_{\mathfrak{q}}^5{\left[4\right]}_{\mathfrak{q}}}\right]\hfill \\ \hfill & +\frac{3}{2}\left[\frac{\left(\mathfrak{q}-1\right){\left[2\right]}_{\mathfrak{q}}^2+1}{{\left[2\right]}_{\mathfrak{q}}^2}-\frac{\mathfrak{q}(2\mathfrak{q}+{\mathfrak{q}}^2){\left[3\right]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^3+1}{{\left[2\right]}_{\mathfrak{q}}{\left[3\right]}_{\mathfrak{q}}}\right.\hfill \\ \hfill & \left.-{\mathfrak{q}}^3\frac{({\left[2\right]}_{\mathfrak{q}}^4-1){\left[3\right]}_{\mathfrak{q}}-(3\mathfrak{q}+3{\mathfrak{q}}^2+{\mathfrak{q}}^3){\left[2\right]}_{\mathfrak{q}}{\left[4\right]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^3{\left[3\right]}_{\mathfrak{q}}{\left[4\right]}_{\mathfrak{q}}}\right].\hfill \end{array}$$

(16)

One can see the validity of the inequality (16) in Figure 1.

Theorem 4.

With the suppositions of Lemma 3, if${\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\right|}^{{\ell }_1}$, ${\ell }_1>1$, is $(\alpha ,m)$-convex function on $[{\xi }_1,{\xi }_2]$, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}{\left(\lambda \right)}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right|\hfill \\ \hfill & \le \frac{{\mathfrak{q}}^3{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}^{\frac{{\ell }_1+\alpha +1}{{\ell }_1}+\frac{2{\ell }_2+1}{{\ell }_2}}{[\alpha +1]}_{\mathfrak{q}}^{\frac{1}{{\ell }_1}}{\left[2{\ell }_2+1\right]}_{\mathfrak{q}}^{\frac{1}{{\ell }_2}}}{\left({\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}+m\left({\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-1\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2{(1-\frac{1}{{\left[2\right]}_{\mathfrak{q}}})}_{\mathfrak{q}}^{\frac{2{\ell }_2+1}{{\ell }_2}}}{{\left[2\right]}_{\mathfrak{q}}^{\frac{{\ell }_1+\alpha +1}{{\ell }_1}}{[\alpha +1]}_{\mathfrak{q}}^{\frac{1}{{\ell }_1}}{\left[2{\ell }_2+1\right]}_{\mathfrak{q}}^{\frac{1}{{\ell }_2}}}\hfill \\ \hfill & \times {\left(({\left[2\right]}_{\mathfrak{q}}^{\alpha +1}-1){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}+m\left(\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}+1\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}},\hfill \end{array}$$

(17)

where $\frac{1}{{\ell }_1}+\frac{1}{{\ell }_2}=1.$

Proof. 

By implementing Hölder’s inequality on Lemma 3, we attain

$$\begin{array}{cc}\hfill & \left|\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}{\left(\lambda \right)}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right|\hfill \\ \hfill & \le \frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}{\mathfrak{q}}^3{\left(\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{\lambda }^{2{\ell }_2}{d}_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_2}}{\left(\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\lambda a+m\left(1-\lambda \right){\xi }_2\right)\right|}^{{\ell }_1}{d}_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}{\left(\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^{2{\ell }_2}{d}_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_2}}{\left(\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\lambda {\xi }_1+m\left(1-\lambda \right){\xi }_2\right)\right|}^{{\ell }_1}{d}_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_1}}.\hfill \end{array}$$

Since ${\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\right|}^{{\ell }_1}$ is $(\alpha ,m)$-convex on $\left[{\xi }_1,{\xi }_2\right]$, we have

$${\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\lambda {\xi }_1+m\left(1-\lambda \right){\xi }_2\right)\right|}^{{\ell }_1}\le {\lambda }^{\alpha }{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}+m\left(1-{\lambda }^{\alpha }\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}.$$

By Lemma 2, we get

$$\begin{array}{cc}\hfill & \left|\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}{\left(\lambda \right)}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right|\hfill \\ \hfill & \le \frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}{\mathfrak{q}}^3{\left(\frac{1}{{\left[2\right]}_{\mathfrak{q}}^{2{\ell }_2+1}{\left[2{\ell }_2+1\right]}_{\mathfrak{q}}}\right)}^{\frac{1}{{\ell }_2}}\hfill \\ \hfill & \times {\left(\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}\left({\lambda }^{\alpha }{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}+m\left(1-{\lambda }^{\alpha }\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)d_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}{\left(\frac{{(1-\frac{1}{{\left[2\right]}_{\mathfrak{q}}})}_{\mathfrak{q}}^{2{\ell }_2+1}}{{\left[2{\ell }_2+1\right]}_{\mathfrak{q}}}\right)}^{\frac{1}{{\ell }_2}}\hfill \\ \hfill & \times {\left(\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}\left({\lambda }^{\alpha }{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}+m\left(1-{\lambda }^{\alpha }\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)d_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & =\left(\frac{{\mathfrak{q}}^3{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}{\left[2\right]}_{\mathfrak{q}}^{\frac{2{\ell }_2+1}{{\ell }_2}}{\left[2{\ell }_2+1\right]}_{\mathfrak{q}}^{\frac{1}{{\ell }_2}}}\right){\left(\frac{{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +1]}_{\mathfrak{q}}}+m\frac{\left({\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-1\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +1]}_{\mathfrak{q}}}\right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}\frac{{(1-\frac{1}{{\left[2\right]}_{\mathfrak{q}}})}_{\mathfrak{q}}^{\frac{2{\ell }_2+1}{{\ell }_2}}}{{\left[2{\ell }_2+1\right]}_{\mathfrak{q}}^{\frac{1}{{\ell }_2}}}\hfill \\ \hfill & \times {\left(\frac{\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +1}-1\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +1]}_{\mathfrak{q}}}+m\frac{\left(\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}+1\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +1]}_{\mathfrak{q}}}\right)}^{\frac{1}{{\ell }_1}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\left(\frac{{\mathfrak{q}}^3{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}^{\frac{{\ell }_1+\alpha +1}{{\ell }_1}}{\left[2\right]}_{\mathfrak{q}}^{\frac{2{\ell }_2+1}{{\ell }_2}}{\left[2{\ell }_2+1\right]}_{\mathfrak{q}}^{\frac{1}{{\ell }_2}}}\right){\left({\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}+\left({\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-1\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2{(1-\frac{1}{{\left[2\right]}_{\mathfrak{q}}})}_{\mathfrak{q}}^{\frac{2{\ell }_2+1}{{\ell }_2}}}{{\left[2\right]}_{\mathfrak{q}}^{\frac{3+{\ell }_1}{{\ell }_1}}{\left[2{\ell }_2+1\right]}_{\mathfrak{q}}^{\frac{1}{{\ell }_2}}}\hfill \\ \hfill & \times {\left(\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +1}-1\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}+m\left(\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}+1\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}}.\hfill \end{array}$$

This completes the proof. □

Remark 5.

Taking limit $\mathfrak{q}\to 1^{-}$ and $\alpha =m=1$ in Theorem 4, we attain

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right)d\lambda -\mathfrak{G}\left(\frac{{\xi }_1+{\xi }_2}{2}\right)\right|\hfill \\ \hfill & \le \frac{{\left({\xi }_2-{\xi }_1\right)}^2}{2^{4+\frac{2}{{\ell }_1}}{\left(2{\ell }_2+1\right)}^{\frac{1}{{\ell }_2}}}\left\{{\left(3{\left|{\mathfrak{G}}^{″}\left({\xi }_1\right)\right|}^{{\ell }_1}+{\left|{\mathfrak{G}}^{″}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}}+{\left({\left|{\mathfrak{G}}^{″}\left({\xi }_1\right)\right|}^{{\ell }_1}+3{\left|{\mathfrak{G}}^{″}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}}\right\},\hfill \end{array}$$

which was given in ([24], Theorem 3).

Remark 6.

By choosing $\alpha =1$ and $m=1$, we recapture ([22], Theorem 4).

Theorem 5.

With the assumptions of Lemma 3, if ${\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\right|}^{{\ell }_1}$, ${\ell }_1\ge 1$, is $(\alpha ,m)$-convex on $\left[{\xi }_1,{\xi }_2\right]$, then we have the inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right){}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right|\hfill \\ \hfill & \le \frac{{\mathfrak{q}}^3{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}^{4-\frac{3}{{\ell }_1}}{\left[3\right]}_{\mathfrak{q}}^{1-\frac{1}{{\ell }_1}}}{\left(\frac{{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +3\right]}_{\mathfrak{q}}}+m\frac{\left({\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +3]}_{\mathfrak{q}}-{\left[3\right]}_{\mathfrak{q}}\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +3\right]}_{\mathfrak{q}}{\left[3\right]}_{\mathfrak{q}}}\right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & +\frac{{\left(\mathfrak{q}+{\mathfrak{q}}^2-{\mathfrak{q}}^3\right)}^{1-\frac{1}{{\ell }_1}}{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}^{4-\frac{3}{{\ell }_1}}{\left[3\right]}_{\mathfrak{q}}^{1-\frac{1}{{\ell }_1}}}\left(\left[\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +2]}_{\mathfrak{q}}-\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha +2}{[\alpha +1]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right.\right.\hfill \\ \hfill & \left.+\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left(\mathfrak{q}{[\alpha +1]}_{\mathfrak{q}}-{[\alpha +2]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}+\frac{{\mathfrak{q}}^3{[\alpha +2]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right]{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}\hfill \\ \hfill & +m\left(\frac{(\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}+1){\left[2\right]}_{\mathfrak{q}}^2{[\alpha +2]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +1\right]}_{\mathfrak{q}}{\left[\alpha +2\right]}_{\mathfrak{q}}}\right.\hfill \\ \hfill & -\frac{\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}{[\alpha +1]}_{\mathfrak{q}}\left((2\mathfrak{q}+{\mathfrak{q}}^2){\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +2]}_{\mathfrak{q}}({\left[2\right]}_{\mathfrak{q}}^{\alpha +2}-1){\left[2\right]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +1\right]}_{\mathfrak{q}}{\left[\alpha +2\right]}_{\mathfrak{q}}}\hfill \\ \hfill & \left.-{\mathfrak{q}}^3\frac{({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1){\left[3\right]}_{\mathfrak{q}}-(3\mathfrak{q}+3{\mathfrak{q}}^2+{\mathfrak{q}}^3){\left[2\right]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[3\right]}_{\mathfrak{q}}{\left[\alpha +3\right]}_{\mathfrak{q}}}\right){\left.{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}}.\hfill \end{array}$$

Proof. 

Using Power-mean inequality on Lemma 3 and then also using $(\alpha ,m)$-convexity of ${\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\right|}^{{\ell }_1}$, we get

$$\begin{array}{cc}\hfill & \left|\frac{1}{m{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{m{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right){}^{m{\xi }_2}{d}_{\mathfrak{q}}\lambda -\mathfrak{G}\left(\frac{{\xi }_1+\mathfrak{q}m{\xi }_2}{{\left[2\right]}_{\mathfrak{q}}}\right)\right|\hfill \\ \hfill & \le \frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}{\mathfrak{q}}^3{\left(\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{\lambda }^2{d}_{\mathfrak{q}}\lambda \right)}^{1-\frac{1}{{\ell }_1}}{\left(\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{\lambda }^2{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\lambda {\xi }_1+m\left(1-\lambda \right){\xi }_2\right)\right|}^{{\ell }_1}{d}_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}{\left(\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{d}_{\mathfrak{q}}\lambda \right)}^{1-\frac{1}{{\ell }_1}}\hfill \\ \hfill & \times {\left(\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left(\lambda {\xi }_1+m\left(1-\lambda \right){\xi }_2\right)\right|}^{{\ell }_1}{d}_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & \le \frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}\frac{{\mathfrak{q}}^3}{{\left[2\right]}_{\mathfrak{q}}^{3-\frac{3}{{\ell }_1}}{\left[3\right]}_{\mathfrak{q}}^{1-\frac{1}{{\ell }_1}}}\hfill \\ \hfill & \times {\left({\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}{\lambda }^{\alpha +2}{d}_{\mathfrak{q}}\lambda +m{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\underset{0}{\overset{1/{\left[2\right]}_{\mathfrak{q}}}{\int }}\left({\lambda }^2-{\lambda }^{\alpha +2}\right)d_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & +\frac{{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}}\frac{{\left[{(1-\frac{1}{{\left[2\right]}_{\mathfrak{q}}})}_{\mathfrak{q}}^3\right]}^{1-\frac{1}{{\ell }_1}}}{{\left[3\right]}_{\mathfrak{q}}^{1-\frac{1}{{\ell }_1}}}\hfill \\ \hfill & \times {\left({\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}{\lambda }^{\alpha }{(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{d}_{\mathfrak{q}}\lambda +m{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\underset{1/{\left[2\right]}_{\mathfrak{q}}}{\overset{1}{\int }}\left(1-{\lambda }^{\alpha }\right){(1-\mathfrak{q}\lambda )}_{\mathfrak{q}}^2{d}_{\mathfrak{q}}\lambda \right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & =\frac{{\mathfrak{q}}^3{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}^{4-\frac{3}{{\ell }_1}}{\left[3\right]}_{\mathfrak{q}}^{1-\frac{1}{{\ell }_1}}}{\left(\frac{{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +3\right]}_{\mathfrak{q}}}+m\frac{\left({\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +3]}_{\mathfrak{q}}-{\left[3\right]}_{\mathfrak{q}}\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +3\right]}_{\mathfrak{q}}{\left[3\right]}_{\mathfrak{q}}}\right)}^{\frac{1}{{\ell }_1}}\hfill \\ \hfill & +\frac{{\left(\mathfrak{q}+{\mathfrak{q}}^2-{\mathfrak{q}}^3\right)}^{1-\frac{1}{{\ell }_1}}{\left(m{\xi }_2-{\xi }_1\right)}^2}{{\left[2\right]}_{\mathfrak{q}}^{4-\frac{3}{{\ell }_1}}{\left[3\right]}_{\mathfrak{q}}^{1-\frac{1}{{\ell }_1}}}\left(\left[\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +1}{[\alpha +2]}_{\mathfrak{q}}-\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha +2}{[\alpha +1]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right.\right.\hfill \\ \hfill & \left.+\frac{{\left[2\right]}_{\mathfrak{q}}^2{[\alpha +3]}_{\mathfrak{q}}\left(\mathfrak{q}{[\alpha +1]}_{\mathfrak{q}}-{[\alpha +2]}_{\mathfrak{q}}\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}+\frac{{\mathfrak{q}}^3{[\alpha +2]}_{\mathfrak{q}}\left({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1\right)}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{[\alpha +1]}_{\mathfrak{q}}{[\alpha +2]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}\right]{\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_1\right)\right|}^{{\ell }_1}\hfill \\ \hfill & +m\left(\frac{(\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +1]}_{\mathfrak{q}}-{\left[2\right]}_{\mathfrak{q}}^{\alpha +1}+1){\left[2\right]}_{\mathfrak{q}}^2{[\alpha +2]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +1\right]}_{\mathfrak{q}}{\left[\alpha +2\right]}_{\mathfrak{q}}}\right.\hfill \\ \hfill & -\frac{\mathfrak{q}{\left[2\right]}_{\mathfrak{q}}{[\alpha +1]}_{\mathfrak{q}}((2\mathfrak{q}+{\mathfrak{q}}^2){\left[2\right]}_{\mathfrak{q}}^{\alpha }{[\alpha +2]}_{\mathfrak{q}}-({\left[2\right]}_{\mathfrak{q}}^{\alpha +2}-1){\left[2\right]}_{\mathfrak{q}})}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[\alpha +1\right]}_{\mathfrak{q}}{\left[\alpha +2\right]}_{\mathfrak{q}}}\hfill \\ \hfill & {\left.\left.-{\mathfrak{q}}^3\frac{({\left[2\right]}_{\mathfrak{q}}^{\alpha +3}-1){\left[3\right]}_{\mathfrak{q}}-(3\mathfrak{q}+3{\mathfrak{q}}^2+{\mathfrak{q}}^3){\left[2\right]}_{\mathfrak{q}}{[\alpha +3]}_{\mathfrak{q}}}{{\left[2\right]}_{\mathfrak{q}}^{\alpha +3}{\left[3\right]}_{\mathfrak{q}}{\left[\alpha +3\right]}_{\mathfrak{q}}}\right){\left|{}^{{\xi }_2}{D}_{\mathfrak{q}}^2\mathfrak{G}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}}.\hfill \end{array}$$

This completes the proof. □

Remark 7.

Choosing limit as $\mathfrak{q}\to 1^{-}$ in Theorem 5, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{{\xi }_2-{\xi }_1}\underset{{\xi }_1}{\overset{{\xi }_2}{\int }}\mathfrak{G}\left(\lambda \right)d\lambda -\mathfrak{G}\left(\frac{{\xi }_1+{\xi }_2}{2}\right)\right|\hfill \\ \hfill & \le \frac{{\left({\xi }_2-{\xi }_1\right)}^2}{3\left(2^{4+\frac{3}{{\ell }_1}}\right)}\left\{{\left(5{\left|{\mathfrak{G}}^{″}\left({\xi }_1\right)\right|}^{{\ell }_1}+3{\left|{\mathfrak{G}}^{″}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}}+{\left(3{\left|{\mathfrak{G}}^{″}\left({\xi }_1\right)\right|}^{{\ell }_1}+5{\left|{\mathfrak{G}}^{″}\left({\xi }_2\right)\right|}^{{\ell }_1}\right)}^{\frac{1}{{\ell }_1}}\right\},\hfill \end{array}$$

which was given in ([24], Theorem 4).

Remark 8.

By choosing $\alpha =1$ and $m=1$, we recapture ([22], Theorem 5).

4. Concluding Remarks

We conclude our paper by saying that there is not much literature present while dealing with quadrature inequalities for twice quantum differentiable convexities, as they are not easy to work on. Thus, in this study, we give new extensions of quantum midpoint inequalities by employing ${}^{{\xi }_2}{D}_q$-derivative and $q^{{\xi }_2}$ -integral under the influence of twice quantum differentiable $(\alpha ,m)$-convex functions. As a result, several fresh estimates of midpoint inequalities are achieved by utilizing quantum Holder and Power mean inequalities. As an example, we provide graphical analysis to explain the correctness of our results. As a special cases for $m=1$ and quantum parameter $q=1$, we recapture results in quantum and classical calculus. This idea may also be expanded to obtain other quadrature inequalities i.e Simpson and Newton type pertaining twice quantum differentiable convexities. One may also think about the case for ${}_{{\xi }_1}{D}_q$-derivative and $q_{{\xi }_1}$-integral. In future, we aim to work on such estimations of quantum midpoint inequalities by employing Mercer’s scheme. It is pertinent to mention that such extensions are quite open to discovery in $(p,q)$-calculus and for co-ordinated convex functions in quantum calculus. A good open problem is to investigate fractional quantum midpoint-type inequalities. We think it is an intriguing and fresh subject for academics, who can produce equivalent inequalities using various convexities.