Some generalized fractional integral Simpson’s type inequalities with applications

Sabir Hussain; Javairiya Khalid; Yu Ming Chu; Sabir Hussain; Javairiya Khalid; Yu Ming Chu

doi:10.3934/math.2020375

AIMS Mathematics

2020, Volume 5, Issue 6: 5859-5883. doi: 10.3934/math.2020375

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Research article

Some generalized fractional integral Simpson’s type inequalities with applications

1.
Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan
2.
Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China
3.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China

Received: 10 June 2020 Accepted: 08 July 2020 Published: 14 July 2020
MSC : 26D15, 26A51

In the article, we establish a Simpson-type generalized identity containing multi-parameters and derive some new estimates for the generalized Simpson's quadrature rule via the Raina fractional integrals. As applications, we provide several inequalities for the f -divergence measures and probability density functions.

Keywords:

Citation: Sabir Hussain, Javairiya Khalid, Yu Ming Chu. Some generalized fractional integral Simpson’s type inequalities with applications[J]. AIMS Mathematics, 2020, 5(6): 5859-5883. doi: 10.3934/math.2020375

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Abstract

1. Introduction

Over few years, the fractional calculus has attracted the attention of many researchers due to its has wide applications in pure and applied mathematics ^{[1,2,3,4,5,6,7]}. Like ordinary calculus, the fractional integral and derivative have not unique representation, with the passage of time, different authors have different representations. It is well-known that inequality is an indispensable research object in mathematics, it can give explicit error bounds for some known and some new quadrature formulae, for example, the Simpson's inequality ^[8], Jensen's inequality ^[9,10], Hermite-Hadamard's inequality ^{[11,12,13,14,15]} and integral inequalities ^{[16,17,18,19,20,21]}. The following inequality is well known as Simpson's inequality which provides an error bound for the Simpson's rule.

Theorem 1.1. (See ^[7]) Let $a, b\in \mathbb{R}$ with $a < b$ , and $f:[a, b]\rightarrow\mathbb{R}$ be a four times differentiable function on $(a, b)$ such that $\|f^{(4)}\|_{\infty} = \sup_{x\in(a, b)}|f^{(4)}|(x) < \infty$ . Then the inequality

$\begin{equation*} \left|\frac{1}{3}\left[\frac{f(a)+f(b)}{2}+2f\left(\frac{a+b}{2}\right)\right]-\frac{1}{b-a}\int_{a}^{b}f(x)dx\right| \leq\frac{\|f^{(4)}\|_ {\infty}(b-a)^{4}}{2880} \end{equation*}$

holds.

In ^[22], Dragomir et al. improved Theorem 1.1 to the following Theorem 1.2.

Theorem 1.1. (See ^[22]) Let $a, b\in \mathbb{R}$ with $a < b$ , and $f:[a, b]\rightarrow\mathbb{R}$ be a differentiable function on $(a, b)$ such that its derivative is continuous on $(a, b)$ and $\|f'\|_{1} = \int_{a}^{b}|f^{\prime}(x)|dx < \infty$ . Then the inequality

$\begin{equation*} \left|\frac{1}{3}\left[\frac{f(a)+f(b)}{2}+2f\left(\frac{a+b}{2}\right)\right]-\frac{1}{b-a}\int_{a}^{b}f(x)dx\right|\leq\frac{b-a}{3}||f'||_{1} \end{equation*}$

holds.

Recently, the Simpson type inequalities have been the subject of intensive research since many important inequalities can be obtained from the Simpson inequality. The main purpose of the article is to provide several generalized fractional integral versions of the Simpson's inequality and give their applications in $f$ -divergence measures and probability density functions.

2. Preliminaries and assumptions

Definition 2.1. (See ^[23]) Let $p\in\mathbb{R}$ with $p\neq 0$ and $I\subseteq(0, \infty)$ be an interval. Then the real-valued function $f: I\rightarrow\mathbb{R}$ is said to be $p$ -convex (concave) if the inequality

$\begin{equation*} f(\sqrt[p]{tx^{p}+(1-t)y^{p}})\leq (\geq) tf(x)+(1-t)f(y) \end{equation*}$

holds for all $x, y\in I$ and $t\in[0, 1]$ . Moreover, if $s\in(0, 1]$ , then the function $f$ is said to be $(s, p)$ -convex (concave) if the inequality

$\begin{equation*} f(\sqrt[p]{tx^{p}+(1-t)y^{p}})\leq (\geq) t^{s}f(x)+(1-t)^{s}f(y) \end{equation*}$

holds for all $x, y\in I$ and $t\in[0, 1]$ .

Let $p > 0$ , $s\in(0, 1]$ and $f: (0, \infty)\rightarrow(0, \infty)$ be defined by $f(x) = x^{sp}$ . Then we clearly see that $f$ is a $(s, p)$ -convex function.

Definition 2.2. (See ^[24]) Let $\alpha > 0$ , $[a, b]\subseteq\mathbb{R}$ be a finite interval and $f: [a, b]\rightarrow\mathbb{R}$ be a real-valued function such that $f\in L[a, b]$ . Then the right-hand side and the left-hand side Riemann-Liouville fractional integrals $\mathcal{J}_{a+}^{\alpha}f$ and $\mathcal{J}_{b-}^{\alpha}f$ of order $\alpha$ are defined by

$\begin{equation} \ \left(\mathcal{J}_{a+}^{\alpha}f\right)(x) = \frac{1}{\Gamma(\alpha)}\int_{a}^{x}(x-t)^{\alpha-1}f(t)dt \quad (x \gt a) \end{equation}$

(2.1)

and

$\begin{equation} \left(\mathcal{J}_{b-}^{\alpha}f\right)(x) = \frac{1}{\Gamma(\alpha)}\int_{x}^{b}(t-x)^{\alpha-1}f(t)dt \quad (x \lt b), \end{equation}$

(2.2)

respectively.

Definition 2.3. The gamma function $\Gamma$ , beta function $\mathbb{B}$ and the hypergeometric function ${}_{2}F_{1}$ are defined by

$\begin{equation*} \Gamma(x) = \int_{0}^{\infty}e^{-t}t^{x}dt \quad (x \gt 0), \end{equation*}$

$\begin{equation*} \mathbb{B}(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} = \int_{0}^{1}t^{x-1}(1-t)^{y-1}dt \quad (x, y \gt 0) \end{equation*}$

and

$\begin{equation*} {}_{2}F_{1}(a, b;c, z) = \frac{1}{\mathbb{B}(b, c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}dt \quad (|z| \lt 1), \end{equation*}$

respectively.

Definition 2.4 (See ^[25]) The hypergeometric function ${}_{2}F_{1}$ can be given by

$\begin{equation*} {}_{2}F_{1}(a, [b, c; y], x) = \frac{1}{\mathbb{B}(b, c-b)}\int_{0}^{y} t^{b-1}(1-t)^{c-b-1}(1-xt)^{-a}dt \end{equation*}$

for $y < 1$ and $Re(c) > Re(b) > 0$ .

Definition 2.5 The incomplete beta function $\mathbb{B}(z; x, y)$ is defined by

$\begin{equation*} \mathbb{B}(z;x, y) = \int_{0}^{z}t^{x-1}(1-t)^{y-1}dt \quad (Re(x) \gt Re(y) \gt 0, 0\leq z \lt 1). \end{equation*}$

Raina ^[26] introduced a class of functions as follows

$\begin{equation} \mathfrak{F}_{\rho, \lambda}^{\sigma}(x) = \mathfrak{F}_{\rho, \lambda}^{\sigma(0), \sigma(1), \cdots}(x) = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)}{\Gamma(\rho k+\lambda)}x^{k}\quad (\rho, \lambda\in\mathbb{R}^{+}, x\in\mathbb{R}), \end{equation}$

(2.3)

where the coefficients $\sigma(k)\in\mathbb{R}^{+}$ , $k\in\mathbb{ N}_{0}$ form a bounded sequence. By using (2.3), in ^[26,27], the authors defined the left-side and right-sided fractional integral operators

$\begin{equation} \left(\mathfrak{J}_{\rho, \lambda, a+;w }^{\sigma}\phi\right)(x) = \int_{a}^{x}(x-t)^{\lambda-1}\mathfrak{F}_{\rho, \lambda}^{\sigma}[w(x-t)^{\rho}]\phi(t)dt \quad (x \gt a) \end{equation}$

(2.4)

and

$\begin{equation} \left(\mathfrak{J}_{\rho, \lambda, b-;w }^{\sigma}\phi\right)(x) = \int_{x}^{b}(t-x)^{\lambda-1}\mathfrak{F}_{\rho, \lambda}^{\sigma}[w(t-x)^{\rho}]\phi(t)dt \quad (x \lt b), \end{equation}$

(2.5)

respectively, where $w\in\mathbb{R}$ and $\phi$ is a function such that the integrals on right hand sides exit. It is easy to verify that $\mathfrak{J}_{\rho, \lambda, a+; w }^{\sigma}\phi(x)$ and $\mathfrak{J}_{\rho, \lambda, b-; w }^{\sigma}\phi(x)$ are bounded integral operators on $L(a, b)$ if $\mathfrak{M} = \mathfrak{F}_{\rho, \lambda+1}^{\sigma}[w(b-a)^{\rho}] < \infty$ . In fact, if $\phi\in L(a, b)$ , then we have

$\begin{equation*} \left\|\mathfrak{J}_{\rho, \lambda, a+;w }^{\sigma}\phi\right\|_{1}\leq \mathfrak{M}(b-a)^{\lambda}\|\phi\|_{1}, \quad \left\|\mathfrak{J}_{\rho, \lambda, b-;w }^{\sigma}\phi\right\|_{1}\leq \mathfrak{M}(b-a)^{\lambda}\|\phi\|_{1}. \end{equation*}$

Let $\lambda\rightarrow\alpha$ , adn $\sigma(0)\rightarrow1$ and $w\rightarrow0$ in (2.4) and (2.5), respectively, then we get (2.1) and (2.2). Before starting our main results, we introduce some notations as follows.

$\begin{equation*} G_{1}(a, b;\xi, \varrho, \varepsilon) = {}_{2}F_{1}\left(\frac{\xi p-\xi}{p}, \beta+\xi\rho k+\varepsilon+1; \xi\beta+\xi\rho k+\varrho+2, \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right), \end{equation*}$

$\begin{equation*} G_{2}(a, b;\xi, \varrho, \varepsilon) = {}_{2}F_{1}\left(\frac{\xi p-\xi}{p}, \left[\xi\beta+\xi\rho k +\varepsilon+1, \xi\beta+\xi\rho k+\varrho+2;\frac{1}{2}\right], \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right), \end{equation*}$

$\begin{equation*} H_{1, \lambda}(a, b;\pi, \varpi) = {}_{2}F_{1}\left(\frac{p-1}{p}, \pi, \varpi, \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right), \end{equation*}$

$\begin{equation*} H_{2, \lambda}(a, b;\pi, \varpi) = {}_{2}F_{1}\left(\frac{p-1}{p}, \left[\pi, \varpi; \frac{1}{2}\right], \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right), \end{equation*}$

$\begin{equation*} \sigma_{1} = \sigma(k)\left\{\left(\lambda|f'(a)|+(1-\lambda)|f'( b)|\right)\mathbb{B}(\beta+\rho k+1, s+1)G_{1}(a, b;1, s, 0)\right. \end{equation*}$

$\begin{equation*} \left.+G_{1}(a, b;1, s, s)|f'( b)|\mathbb{B}(\beta+\rho k+s+1, 1)\right\}, \end{equation*}$

$\begin{equation*} \sigma_{2} = \left\{\left[G_{2}(a, b;x, 0, 0)\right]^{\frac{1}{x}}\left[(2^{s+1}(1-\lambda)+\lambda)|f'(b)|^{y}+\lambda (2^{s+1}-1)|f'(a)|^{y}\right]^{\frac{1}{y}}\right. \end{equation*}$

$\begin{equation*} \left.+\left[G_{1}(a, b;x, 0, 0)-G_{2}(a, b;x, 0, 0)\right]^{\frac{1}{x}}\left[(2^{s+1}-\lambda) |f'(b)|^{y}+\lambda|f'(a)|^{y}\right]^{\frac{1}{y}}\right\} \end{equation*}$

$\begin{equation*} \times\sigma(k)\frac{[\mathbb{B}(x\beta+x\rho k+1, 1)]^{\frac{1}{x}}}{[2^{s+1}(s+1)]^{\frac{1}{y}}}, \end{equation*}$

$\begin{equation*} \sigma_{3} = \sigma(k)[\mathbb{B}(\beta+\rho k+1, 1)]^{\frac{x-1}{x}}\left\{ [G_{2}(a, b;1, 0, 0)]^{\frac{x-1}{x}} \left[\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}\right.\right. \end{equation*}$

$\begin{equation*} \left.\times\mathbb{B}(\beta+\rho k+1, s+1)G_{2}(a, b;1, s, 0)+G_{2}(a, b;1, s, s)|f'( b)|^{x}\mathbb{B}(\beta+\rho k+s+1, 1)\right]^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +[G_{1}(a, b;1, 0, 0)-G_{2}(a, b;1, 0, 0)]^{\frac{x-1}{x}}\left[\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}\mathbb{B}(\beta+\rho k+1, s+1)\right. \end{equation*}$

$\begin{equation*} \times\{G_{1}(a, b;1, s, 0)-G_{2}(a, b;1, s, 0)\}+|f'( b)|^{x}\{G_{1}(a, b;1, s, s)-G_{2}(a, b;1, s, s)\}\ \end{equation*}$

$\begin{equation*} \left.\left.\times\mathbb{B}(\beta+\rho k+s+1, 1)\right]^{\frac{1}{x}}\right\}. \end{equation*}$

3. Main results

To establish our results for generalized Simpson's type inequality using $(s, p)$ -convex function, we need the following lemma.

Lemma 3.1. Let $I\subseteq\mathbb{R}^{+}$ be an interval, $I^{\circ}$ be the interval of $I$ , $a, b\in I$ with $a < b$ , $\rho, \beta > 0$ , $p\in\mathbb{R}$ with $p\neq 0$ , and $g(\xi) = \sqrt[p]{\xi}$ for $\xi > 0$ . Then the identity

$\begin{equation*} \psi(t, a, b;f) = p\left[\lambda(b^{p}- a^{p})\right]^{\beta}\left[\left\{2^{-\beta} \mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p}- a^{p})}{4}\right)^{\rho}\right]-u\right\}\right. \end{equation*}$

$\begin{equation*} \left.\times f\left(\sqrt[p]{\frac{\lambda a^{p}+(2-\lambda)b^{p}}{2}}\right)+uf\left(\sqrt[p]{\lambda a^{p} +(1-\lambda)b^{p}}\right)\right]-p\left(\mathfrak{J}_{\rho, \beta, \left[\frac{\lambda a^{p} +(2-\lambda)b^{p}}{2}\;\;\;\;\right]^{-};\frac{w}{2^{\rho}}}^{\sigma}f{\circ}g\right) \end{equation*}$

$\begin{equation*} \times(\lambda a^{p}+(1-\lambda)b^{p})+p\left[\lambda(b^{p}- a^{p})\right]^{\beta} \left[\left\{\mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho}\right]-v\right\}f(b)\right. \end{equation*}$

$\begin{equation*} \left.-\left\{2^{-\beta} \mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p}- a^{p})}{4}\right)^{\rho}\right]-v\right\} f\left(\sqrt[p]{\frac{\lambda a^{p}+(2-\lambda)b^{p}}{2}}\right)\right] -p\left\{\left(\mathfrak{J}_{\rho, \beta, {b^{p}}^{-};\frac{w}{2^{\rho}}}^{\sigma}f{\circ}g\right)\right. \end{equation*}$

$\begin{equation*} \left.\times(\lambda a^{p}+(1-\lambda)b^{p})-\left(\mathfrak{J}_{\rho, \beta, \left[\frac{\lambda a^{p} +(2-\lambda)b^{p}}{2}\;\;\;\;\right]^{-};\frac{w}{2^{\rho}}}^{\sigma}f{\circ}g\right)\left(\lambda a^{p}+(1-\lambda)b^{p}\right)\right\} \end{equation*}$

$\begin{equation*} = [\lambda(b^{p}- a^{p})]^{1+\beta}\int_{0}^{\frac{1}{2}}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]-u\right\} \left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times f'\left(\sqrt[p]{(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}}\right)dt+[\lambda(b^{p}- a^{p})]^{1+\beta} \end{equation*}$

$\begin{equation*} \times\int_{\frac{1}{2}}^{1}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]-v\right\} [(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation} \times f'\left(\sqrt[p]{(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}}\right)dt. \end{equation}$

(3.1)

holds for $u, w\in\mathbb{R}$ and $\lambda\in[0, 1]$ .

Proof. Integrating by parts leads to

$\begin{equation*} I_{1} = \int_{0}^{\frac{1}{2}}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p} -a^{p})}{2}\right)^{\rho}t^{\rho}\right]-u\right\}[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times f'\left(\sqrt[p]{(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}}\right)dt \end{equation*}$

$\begin{equation*} = \bigg|\frac{pt^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho} t^{\rho}\right]-pu}{\lambda(b^{p}-a^{p})}f\left(\sqrt[p]{(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}}\right)\bigg|_{0}^{\frac{1}{2}} \end{equation*}$

$\begin{equation*} -\int_{0}^{\frac{1}{2}}\frac{pt^{\beta-1}\;\;\mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p} -a^{p})}{2}\right)^{\rho}t^{\rho}\right]}{\lambda(b^{p}- a^{p})} f\left(\sqrt[p]{(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}}\right)dt \end{equation*}$

$\begin{equation*} = \frac{p}{\lambda(b^{p}-a^{p})}\left[\left\{2^{-\beta} \mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p}-a^{p})}{4}\right)^{\rho}\right]-u\right\} f\left(\sqrt[p]{\frac{\lambda a^{p}+(2-\lambda)b^{p}}{2}}\right)\right. \end{equation*}$

$\begin{equation*} \left.+uf\left(\sqrt[p]{(\lambda a^{p}+(1-\lambda)b^{p})}\right)\right] -\int_{0}^{\frac{1}{2}}\frac{pt^{\beta-1}\;\;\mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p} -a^{p})}{2}\right)^{\rho}t^{\rho}\right]}{\lambda(b^{p}- a^{p})} \end{equation*}$

$\begin{equation*} \times f\left(\sqrt[p]{(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}}\right)dt. \end{equation*}$

Let $x = (1-t)\left(\lambda a^{p}+(1-\lambda)b^{p}\right)+t b^{p}$ . Then $dx = [\lambda(b^{p}-a^{p})]dt$ , $0\leq t\leq1/2$ is equivalent to $(\lambda a^{p}+(1-\lambda)b^{p})\leq x\leq \frac{\lambda a^{p}+(2-\lambda)b^{p}}{2}$ . Therefore, we get

$\begin{equation*} I_{1} = \frac{p}{\lambda(b^{p}- a^{p})}\left[\left\{2^{-\beta} \mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p}- a^{p})}{4}\right)^{\rho}\right]-u\right\} f\left(\sqrt[p]{\frac{\lambda a^{p}+(2-\lambda)b^{p}}{2}}\right)\right. \end{equation*}$

$\begin{equation*} \left.+uf\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]-\frac{p}{[\lambda(b^{p}- a^{p})]^{1+\beta}}\int_{\lambda a^{p}+(1-\lambda)b^{p}}^{\frac{\lambda a^{p}+(2-\lambda)b^{p}}{2}}[x-(\lambda a^{p}+(1-\lambda)b^{p})]^{\beta-1} \end{equation*}$

$\begin{equation*} \times \mathfrak{F}_{\rho, \beta}^{\sigma} \left[w\left(\frac{x-(\lambda a^{p}+(1-\lambda)b^{p})}{2}\right)^{\rho}\right](f{\circ}g)(x)dx \end{equation*}$

and

$\begin{equation*} \frac{[\lambda(b^{p}- a^{p})]^{1+\beta}}{p}I_{1} = [\lambda(b^{p}- a^{p})]^{\beta}\left[\left\{2^{-\beta} \mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p}- a^{p})}{4}\right)^{\rho}\right]-u\right\}\right. \end{equation*}$

$\begin{equation*} \left.\times f\left(\sqrt[p]{\frac{\lambda a^{p}+(2-\lambda)b^{p}}{2}}\right)+uf\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right] \end{equation*}$

$\begin{equation} -\left(\mathfrak{J}_{\rho, \beta, \left[\frac{\lambda a^{p}+(2-\lambda)b^{p}}{2}\;\;\;\;\right]^{-};\frac{w}{2^{\rho}}}^{\sigma} f{\circ}g\right)(\lambda a^{p}+(1-\lambda)b^{p}). \end{equation}$

(3.2)

Again integrating by parts gives

$\begin{equation*} I_{2} = \int_{\frac{1}{2}}^{1}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]-v\right\} [(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times f'\left(\sqrt[p]{(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}}\right)dt \end{equation*}$

$\begin{equation*} = \bigg| \frac{pt^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]-pv} {\lambda(b^{p}- a^{p})}f\left(\sqrt[p]{(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}}\right)\bigg|_{\frac{1}{2}}^{1} \end{equation*}$

$\begin{equation*} -\int_{\frac{1}{2}}^{1}\frac{pt^{\beta-1}\;\;\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]}{\lambda(b^{p}- a^{p})} f\left(\sqrt[p]{(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}}\right)dt \end{equation*}$

$\begin{equation*} = \frac{p}{\lambda(b^{p}- a^{p})}\left[\left\{\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho}\right]-v\right\}f(b)\right. \end{equation*}$

Let $y = (1-t)\left(\lambda a^{p}+(1-\lambda)b^{p}\right)+t b^{p}$ . Then one has

$\begin{equation*} I_{2} = \frac{p}{\lambda(b^{p}- a^{p})}\left[\left\{\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho}\right]-v\right\}f(b)\right. \end{equation*}$

$\begin{equation*} -\frac{p}{[\lambda(b^{p}- a^{p})]^{1+\beta}}\int_{\frac{\lambda a^\;{p}+(2-\lambda)b^\;{p}}{2}}^{b^\;{p}}\;\;\;[y-(\lambda a^{p}+(1-\lambda)b^{p})]^{\beta-1} \end{equation*}$

$\begin{equation*} \times\mathfrak{F}_{\rho, \beta}^{\sigma}\left[w\left(\frac{y-(\lambda a^{p} +(1-\lambda)b^{p})}{2}\right)^{\rho}\right](f{\circ}g)(y)dy \end{equation*}$

$\begin{equation*} \left.-\int_{\lambda a^{p}+(1-\lambda)b^{p}}^{\frac{\lambda a^{p}+(2-\lambda)b^{p}}{2}}[y-(\lambda a^{p}+(1-\lambda)b^{p})]^{\beta-1}\right] \end{equation*}$

$\begin{equation*} \times \mathfrak{F}_{\rho, \beta}^{\sigma}\left[w\left(\frac{y-(\lambda a^{p} +(1-\lambda)b^{p})}{2}\right)^{\rho}\right](f{\circ}g)(y)dy \end{equation*}$

and

$\begin{equation*} \frac{[\lambda(b^{p}- a^{p})]^{1+\beta}}{p}I_{2} = [\lambda(b^{p}- a^{p})]^{\beta} \left[\left\{\mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[w\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho}\right]-v\right\}f(b)\right. \end{equation*}$

$\begin{equation} -\left\{\left(\mathfrak{J}_{\rho, \beta, {b^{p}}^{-};\frac{w}{2^{\rho}}}^{\sigma} f{\circ}g\right)(\lambda a^{p}+(1-\lambda)b^{p})-\left(\mathfrak{J}_{\rho, \beta, \left[\frac{\lambda a^{p} +(2-\lambda)b^{p}}{2}\;\;\;\;\right]^{-};\frac{w}{2^{\rho}}}^{\sigma}f{\circ}g\right)(\lambda a^{p}+(1-\lambda)b^{p})\right\}. \end{equation}$

(3.3)

Therefore, the desired inequality (3.1) can be obtained by adding (3.2) and (3.3).

Remark 3.1. From Lemma 3.1 we clearly see that

(1) Lemma 3.1 reduces to Lemma 1 of ^[8] if $\beta, \lambda, \sigma(0)\rightarrow1$ , $w\rightarrow0$ , $u\rightarrow\frac{1}{6}$ and $v\rightarrow\frac{5}{6}$ ;

(2) Lemma 3.1 leads to Lemma 2 of ^[8] if $p, \beta, \lambda, \sigma(0)\rightarrow1$ , $w\rightarrow0$ and $a\rightarrow am$ ;

(3) Lemma 3.1 becomes Lemma 2.1 of ^[28] if $p, \beta, \lambda, \sigma(0)\rightarrow1$ , $w\rightarrow0$ ; $u\rightarrow\frac{1}{6}$ , $v\rightarrow\frac{5}{6}$ and $a\rightarrow am$ .

(4) Lemma 3.1 degenerates into Lemma 3 of ^[8] if $\lambda, \sigma(0)\rightarrow1$ , $w\rightarrow0$ and $p > 0$ .

Theorem 3.1. Let $I\subseteq\mathbb{R}^{+}$ be an interval and $I^{\circ}$ be the interior of $I$ , $a, b\in I^{\circ}$ with $a < b$ , $\rho, \beta > 0$ , $u, w\in \mathbb{R}$ , $(s, \lambda)\in(0, 1]\times[0, 1]$ , $g(\xi) = \sqrt[p]{\xi}$ for $\xi > 0$ , and $f:I\rightarrow\mathbb{R}$ be a differentiable function on $I^{\circ}$ such that $|f'|$ is $(s, p)$ -convex. Then one has

$\begin{equation*} |\psi(t, a, b;f)|\leq\frac{[\lambda(b^{p}-a^{p})]^{1+\beta}}{\left(\sqrt[p]{(\lambda a^{p}+(1-\lambda)b^{p})}\right)^{p-1}} \left\{\mathfrak{F}_{\rho, \beta+1}^{\sigma_{1}}\left[|w|\left(\frac{\lambda(b^{p}-a^{p})}{2}\right)^{\rho}\right]\right. \end{equation*}$

$\begin{equation*} +\frac{(\lambda|f'(a)|+(1-\lambda)|f'( b)|)\{(|u|-|v|)H_{2, \lambda}(a, b;1, s+2)+|v|H_{1, \lambda}(a, b;1, s+2)\}}{s+1} \end{equation*}$

$\begin{equation} \left.+\frac{|f'( b)|\{(|u|-|v|)H_{2, \lambda}(a, b;s+1, s+2)+|v|H_{1, \lambda}(a, b;s+1, s+2)\}}{s+1}\right\}. \end{equation}$

(3.4)

Proof. It follows from (3.1) and the $(s, p)$ -convexity of $|f'|$ that

$\begin{equation} \left|\psi(t, a, b;f)\right|\leq [\lambda (b^{p}-a^{p})]^{1+\beta}[|I_{1}|+|I_{2}|], \end{equation}$

(3.5)

$\begin{equation*} |I_{1}| = \bigg|\int_{0}^{\frac{1}{2}}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]-u\right\} [(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times f'\left(\left((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right)^{\frac{1}{p}}\right)dt\bigg| \end{equation*}$

$\begin{equation*} \leq\int_{0}^{\frac{1}{2}}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[|w|\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right] +|u|\right\}[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\bigg|f'\left(((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p})^{\frac{1}{p}}\right)\bigg|dt \end{equation*}$

$\begin{equation*} = \int_{0}^{\frac{1}{2}}\left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k} \left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k}}{\Gamma(\rho k+\beta+1)}t^{\beta+\rho k}+|u|\right\} \end{equation*}$

$\begin{equation*} \times[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \bigg|f'\left(\left((1-t)(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}})^{p}+t b^{p}\right)^{\frac{1}{p}}\right)\bigg|dt \end{equation*}$

$\begin{equation*} \leq\int_{0}^{\frac{1}{2}}\left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k} \left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k} }{\Gamma(\rho k+\beta+1)}t^{\beta+\rho k}+|u|\right\} \end{equation*}$

$\begin{equation*} \times\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{1-p} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\left[(1-t)^{s}\{\lambda|f'(a)|+(1-\lambda)|f'( b)|\}+t^{s}|f'( b)|\right]dt \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k} }{\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p-1}\Gamma(\rho k+\beta+1)}\left\{\left(\lambda|f'(a)|\right.\right. \end{equation*}$

$\begin{equation*} \left.+(1-\lambda)|f'( b)|\right)\int_{0}^{\frac{1}{2}}t^{\beta+\rho k} (1-t)^{s}\left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt \end{equation*}$

$\begin{equation*} \left.+|f'( b)|\int_{0}^{\frac{1}{2}}t^{\beta+\rho k+s} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}} dt\right\} \end{equation*}$

$\begin{equation*} +|u|\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{1-p}\left\{(\lambda|f'(a)|+(1-\lambda)|f'( b)|)\right. \end{equation*}$

$\begin{equation*} \times\int_{0}^{\frac{1}{2}}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}(1-t)^{s}dt \end{equation*}$

$\begin{equation*} \left.+|f'( b)|\int_{0}^{\frac{1}{2}}t^{s} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p} +(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\} \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}-a^{p})}{2}\right)^{\rho k}} {\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p-1}\Gamma(\rho k+\beta+1)} \end{equation*}$

$\begin{equation*} \times\left\{(\lambda|f'(a)|+(1-\lambda)|f'( b)|)\mathbb{B}(\beta+\rho k+1, s+1)\right. \end{equation*}$

$\begin{equation*} \times{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+1, s+\beta+\rho k+2; \frac{1}{2}\right], \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right) \end{equation*}$

$\begin{equation*} +{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+s+1, \beta+\rho k+s+2; \frac{1}{2}\right], \frac{\lambda (a^{p}- b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right) \end{equation*}$

$\begin{equation*} \left.\times|f'( b)|\mathbb{B}\left(\beta+\rho k+s+1, 1\right)\right\} +|u|\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{1-p} \end{equation*}$

$\begin{equation*} \times\{(\lambda|f'(a)|+(1-\lambda)|f'( b)|)\mathbb{B}(1, s+1) \end{equation*}$

$\begin{equation*} \times{}_{2}F_{1}\left(\frac{p-1}{p}, \left[1, s+2;\frac{1}{2}\right], \frac{\lambda (a^{p}- b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right) \end{equation*}$

$\begin{equation} \left.+|f'( b)|\mathbb{B}(s+1, 1){}_{2}F_{1}\left(\frac{p-1}{p}, \left[s+1, s+2; \frac{1}{2}\right], \frac{\lambda( a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right\} \end{equation}$

(3.6)

and

$\begin{equation*} |I_{2}| = \bigg|\int_{\frac{1}{2}}^{1}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]-v\right\} [(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times f'\left(((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p})^{\frac{1}{p}}\right)dt\bigg| \end{equation*}$

$\begin{equation*} \leq\int_{\frac{1}{2}}^{1}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[|w|\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]+|v|\right\} [(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\bigg|f'\left(\left((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right)^{\frac{1}{p}}\right)\bigg|dt \end{equation*}$

$\begin{equation*} = \int_{\frac{1}{2}}^{1} \left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k} \left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)}t^{\beta+\rho k}+|v|\right\} \end{equation*}$

$\begin{equation*} \times[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\bigg|f'\left(((1-t)(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}})^{p}+t b^{p})^{\frac{1}{p}}\right)\bigg|dt \end{equation*}$

$\begin{equation*} \leq\int_{\frac{1}{2}}^{1} \left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k} \left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k}}{\Gamma(\rho k+\beta+1)}t^{\beta+\rho k}+|v|\right\} \end{equation*}$

$\begin{equation*} \times\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{1-p} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\left[(1-t)^{s}\{\lambda|f'(a)|+(1-\lambda)|f'( b)|\}+t^{s}|f'( b)|\right]dt \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}-a^{p})}{2}\right)^{\rho k}} {\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p-1}\Gamma(\rho k+\beta+1)}\{(\lambda|f'(a)| \end{equation*}$

$\begin{equation*} +(1-\lambda)|f'(b)|)\int_{\frac{1}{2}}^{1}t^{\beta+\rho k}(1-t)^{s} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt \end{equation*}$

$\begin{equation*} \left.+|f'( b)|\int_{\frac{1}{2}}^{1}t^{\beta+\rho k+s} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p} +(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}} dt\right\} \end{equation*}$

$\begin{equation*} +|v|\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{1-p}\{(\lambda|f'(a)|+(1-\lambda)|f'( b)|) \end{equation*}$

$\begin{equation*} \times\int_{\frac{1}{2}}^{1}\left [1-t\frac{\lambda(a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}(1-t)^{s}dt \end{equation*}$

$\begin{equation*} \left.+|f'( b)|\int_{\frac{1}{2}}^{1}t^{s}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}\ \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda( b^{p}-a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\;\right)\;^{p-1}}\;\left\{\mathbb{B}(\beta+\rho k+1, s+1)\right. \end{equation*}$

$\begin{equation*} \times\left[{}_{2}F_{1}\left(\frac{p-1}{p}, \beta+\rho k+1;s+\beta+\rho k+2, \frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right)\nonumber\right. \end{equation*}$

$\begin{equation*} \left.-{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+1, s+\beta+\rho k+2;\frac{1}{2}\right], \frac{\lambda(a^{p}- b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right] \end{equation*}$

$\begin{equation*} \times(\lambda|f'(a)|+(1-\lambda)|f'( b)|)+ |f'( b)|\mathbb{B}(\beta+\rho k+s+1, 1) \end{equation*}$

$\begin{equation*} \times\left[{}_{2}F_{1}\left(\frac{p-1}{p}, \beta+\rho k+s+1, \beta+\rho k+s+2, \frac{\lambda(a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right. \end{equation*}$

$\begin{equation*} \left.\left.-{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+s+1, \beta+\rho k+s+2; \frac{1}{2}\right], \frac{\lambda( a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]\right\} \end{equation*}$

$\begin{equation*} +\frac{|v|}{\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p-1}}\left\{(\lambda|f'(a)|+(1-\lambda)|f'( b)|)\right. \end{equation*}$

$\begin{equation*} \times\mathbb{B}(1, s+1)\left[{}_{2}F_{1}\left(\frac{p-1}{p}, 1, s+2, \frac{\lambda( a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right. \end{equation*}$

$\begin{equation*} \left.-{}_{2}F_{1}\left(\frac{p-1}{p}, \left[1, s+2;\frac{1}{2}\right], \frac{\lambda(a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]+|f'(b)| \end{equation*}$

$\begin{equation*} \times\mathbb{B}(s+1, 1)\left[{}_{2}F_{1}\left(\frac{p-1}{p}, s+1, s+2, \frac{\lambda( a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right. \end{equation*}$

$\begin{equation} \left.\left.-{}_{2}F_{1}\left(\frac{p-1}{p}, \left[s+1, s+2;\frac{1}{2}\right], \frac{\lambda( a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]\right\}. \end{equation}$

(3.7)

Therefore, inequality (3.4) follows easily from (3.5)–(3.7).

Corollary 3.1. Let $I\subseteq\mathbb{R}^{+}$ be an interval and $I^{\circ}$ be the interior of $I$ , $a, b\in I^{\circ}$ with $a < b$ , $\beta > 0$ , $u, w\in \mathbb{R}^{+}$ , $s\in(0, 1]$ , $p < 0$ , $g(\xi) = \sqrt[p]{\xi}$ for $\xi > 0$ , and $f:I\rightarrow\mathbb{R}$ be a differentiable function on $I^{\circ}$ such that $|f'|$ is $(s, p)$ -convex. Then one has

$\begin{equation*} \left|uf(a)+(1-v)f(b)+(v-u)f\left(\sqrt[p]{\frac{a^{p}+b^{p}}{2}}\right)-\frac{1}{b^{p}-a^{p}}\int_{a^{p}}^{b^{p}}(f{\circ}g)(x)dx\right| \end{equation*}$

$\begin{equation*} \leq\frac{b^{p}-a^{p}}{p(s+1)a^{p-1}}\left[(u-v)\left\{ |f'(a)|{}_{2}F_{1}\left(\frac{p-1}{p}, \left[1, s+2; \frac{1}{2}\right], \frac{a^{p}-b^{p}}{a^{p}}\right)\right.\right. \end{equation*}$

$\begin{equation*} \left.+|f'( b)|{}_{2}F_{1}\left(\frac{p-1}{p}, \left[s+1, s+2;\frac{1}{2}\right], \frac{a^{p}-b^{p}}{a^{p}}\right)\right\}+v\left\{|f'(a)|\right. \end{equation*}$

$\begin{equation} \left.\left.\times{}_{2}F_{1}\left(\frac{p-1}{p}, 1, s+2, \frac{a^{p}-b^{p}}{a^{p}}\right) +|f'(b)|{}_{2}F_{1}\left(\frac{p-1}{p}, s+1, s+2, \frac{a^{p}-b^{p}}{a^{p}}\right)\right\}\right]. \end{equation}$

(3.8)

Proof. Let $\beta, \lambda, \sigma(0)\rightarrow1$ and $w = 0$ . Then Corollary 3.1 follows directly from Theorem 3.1.

Theorem 3.2. Let $I\subseteq\mathbb{R}^{+}$ be an interval and $I^{\circ}$ be the interior of $I$ , $a, b\in I^{\circ}$ with $a < b$ , $\rho, \beta > 0$ , $(s, \lambda)\in(0, 1]\times[0, 1]$ , $p\in\mathbb{R}$ with $p\neq 0$ , $u, w, \in \mathbb{R}$ , $y > 1$ , $x = y/(y-1)$ , $g(\xi) = \sqrt[p]{\xi}$ for $\xi > 0$ , and $f:I\rightarrow\mathbb{R}$ be a differentiable function on $I^{\circ}$ such that $|f'|^{y}$ is $(s, p)$ -convex. Then the inequality

$\begin{equation*} |\psi(t, a, b;f)|\leq \frac{[\lambda(b^{p}-a^{p})]^{1+\beta}}{\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{x(p-1)}} \left\{\mathfrak{F}_{\rho, \beta+1}^{\sigma_{2}}\left[|w|\left(\frac{\lambda(b^{p}-a^{p})}{2}\right)^{\rho}\right]\right. \end{equation*}$

$\begin{equation*} +\frac{\left[\frac{\lambda a^{p}+(1-\lambda)b^{p}}{\lambda( a^{p}-b^{p})}\right]^{\frac{1}{x}}} {[2^{s+1}(s+1)]^{\frac{1}{y}}}\left\{|u|\left[\mathbb{B}\left(\frac{\lambda (a^{p}-b^{p})} {2(\lambda a^{p}+(1-\lambda)b^{p})};1, x\frac{p-1}{p}+1\right)\right]^{\frac{1}{x}}\right. \end{equation*}$

$\begin{equation*} \times\left\{(2^{s+1}(1-\lambda)+\lambda)|f'(b)|^{y}+\lambda (2^{s+1}-1)|f'(a)|^{y}\right\}^{\frac{1}{y}} \end{equation*}$

$\begin{equation*} +|v|\left[\mathbb{B}\left(\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}};1, x\frac{p-1}{p}+1\right) -\mathbb{B}\left(\frac{\lambda (a^{p}-b^{p})}{2(\lambda a^{p}+(1-\lambda)b^{p})};1, x\frac{p-1}{p}+1\right)\right]^{\frac{1}{x}} \end{equation*}$

$\begin{equation} \left.\left.\times\left\{(2^{s+1}-\lambda)|f'(b)|^{y}+\lambda|f'(a)|^{y}\right\}^{\frac{1}{y}}\right\}\right\} \end{equation}$

(3.9)

holds.

Proof. It follows from the $(s, p)$ -convexity of $|f'|^{y}$ and Hölder inequality that

$\begin{equation*} |I_{1}| = \left|\int_{0}^{\frac{1}{2}}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]-u\right\} [(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} \left.\times f'\left(((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p})^{\frac{1}{p}}\right)dt\right| \end{equation*}$

$\begin{equation*} \leq\int_{0}^{\frac{1}{2}}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[|w| \left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]+|u|\right\} [(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\left|f'\left(\left((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right)^{\frac{1}{p}}\right)\right|dt \end{equation*}$

$\begin{equation*} = \int_{0}^{\frac{1}{2}}\left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k} \left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho k} }{\Gamma(\rho k+\beta+1)}t^{\beta+\rho k}+|u|\right\} \end{equation*}$

$\begin{equation*} \times\left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\left|f'\left(\left((1-t)(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}})^{p}+t b^{p}\right)^{\frac{1}{p}}\right)\right|dt \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}-a^{p})}{2}\right)^{\rho k}}{\Gamma(\rho k+\beta+1)} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}t^{x(\beta+\rho k)} \left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{x(1-p)}{p}}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}\left|f'\left(\left((1-t)(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}})^{p} +tb^{p}\right)^{\frac{1}{p}}\right)\right|^{y}dt\right\}^{\frac{1}{y}} \end{equation*}$

$\begin{equation*} +|u|\left\{\int_{0}^{\frac{1}{2}}\left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{x(1-p)}{p}}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}\left|f'\left(\left((1-t)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p} +tb^{p}\right)^{\frac{1}{p}}\right)\right|^{y}dt\right\}^{\frac{1}{y}} \end{equation*}$

$\begin{equation*} = \left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}-a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{x(p-1)}}\right. \end{equation*}$

$\begin{equation*} \times\left[\int_{0}^{\frac{1}{2}}t^{x(\beta+\rho k)} \left[1-t\frac{\lambda(a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{x(1-p)}{p}}dt\right]^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +\frac{|u|}{\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{x(p-1)}} \end{equation*}$

$\begin{equation*} \left.\times\left[\int_{0}^{\frac{1}{2}} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p} +(1-\lambda)b^{p}}\right]^{\frac{x(1-p)}{p}}dt\right]^{\frac{1}{x}}\right\} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}[(1-t)^{s}\{\lambda|f'(a)|^{y}+(1-\lambda)|f'( b)|^{y}\}+t^{s}|f'( b)|^{\nu}]dt\right\}^{\frac{1}{y}} \end{equation*}$

$\begin{equation*} = \left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda(b^{p}-a^{p})}{2}\right)^{\rho k}}{\Gamma(\rho k+\beta+1) \left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{x(p-1)}}\sqrt[x]{\mathbb{B}(x\beta+x\rho k+1, 1)}\right. \end{equation*}$

$\begin{equation*} \times\left[{}_{2}F_{1}\left(x\frac{p-1}{p}, \left[x\beta+x\rho k+1, x\beta+x\rho k+2;\frac{1}{2}\right], \frac{\lambda(a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +\frac{|u|}{\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{x(p-1)}} \end{equation*}$

$\begin{equation*} \left.\times\left[\frac{\lambda a^{p}+(1-\lambda)b^{p}}{\lambda (a^{p}-b^{p})}\mathbb{B} \left(\frac{\lambda (a^{p}-b^{p})}{2(\lambda a^{p}+(1-\lambda)b^{p})};1, x\frac{p-1}{p}+1\right)\right]^{\frac{1}{x}}\right\} \end{equation*}$

$\begin{equation} \times\left\{\frac{2^{s+1}(1-\lambda)+\lambda}{2^{s+1}(s+1)} |f'(b)|^{y}+\frac{\lambda (2^{s+1}-1)}{2^{s+1}(s+1)}|f'(a)|^{y}\right\}^{\frac{1}{y}} \end{equation}$

(3.10)

and

$\begin{equation*} |I_{2}| = \left|\int_{\frac{1}{2}}^{1}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma} \left[w\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]-v\right\} [(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} \left.\times f'\left(((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p})^{\frac{1}{p}}\right)dt\right| \end{equation*}$

$\begin{equation*} \times\left|f'\left(((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p})^{\frac{1}{p}}\right)\right|dt \end{equation*}$

$\begin{equation*} = \int_{\frac{1}{2}}^{1}\left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)}t^{\beta+\rho k}+|v|\right\} \end{equation*}$

$\begin{equation*} \times[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\left|f'\left(\left((1-t)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p}+t b^{p}\right)^{\frac{1}{p}}\right)\right|dt \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda(b^{p}-a^{p})}{2}\right)^{\rho k}}{\Gamma(\rho k+\beta+1)} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}t^{x(\beta+\rho k)} \left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{x(1-p)}{p}}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}\left|f'\left(\left((1-t)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p} +tb^{p}\right)^{\frac{1}{p}}\right)\right|^{y}dt\right\}^{\frac{1}{y}} \end{equation*}$

$\begin{equation*} +|v|\left\{\int_{\frac{1}{2}}^{1} \left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{x(1-p)}{p}}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} = \left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{x(p-1)}}\right. \end{equation*}$

$\begin{equation*} \times\left[\int_{\frac{1}{2}}^{1}t^{x(\beta+\rho k)}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{x(1-p)}{p}}dt\right]^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +\frac{|v|}{\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{x(p-1)}} \left.\left[\int_{\frac{1}{2}}^{1}\left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p} +(1-\lambda)b^{p}}\right]^{\frac{x(1-p)}{p}}dt\right]^{\frac{1}{x}}\right\} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}\left[(1-t)^{s}\{\lambda|f'(a)|^{y}+(1-\lambda)|f'( b)|^{y}\}+t^{s}|f'( b)|^{y}\right]dt\right\}^{\frac{1}{y}} \end{equation*}$

$\begin{equation*} = \left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda(b^{p}-a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{x(p-1)}}\sqrt[x]{\mathbb{B}(x\beta+x\rho k+1, 1)}\right. \end{equation*}$

$\begin{equation*} \times\left[{}_{2}F_{1}\left(x\frac{p-1}{p}, \beta+x\rho k+1, x\beta+x\rho k+2, \frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right)\right. \end{equation*}$

$\begin{equation*} \left.-{}_{2}F_{1}\left(x\frac{p-1}{p}, \left[x\beta+x\rho k+1, x\beta+x\rho k+2;\frac{1}{2}\right], \frac{\lambda( a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +\frac{|v|}{\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{x(p-1)}}\left(\frac{\lambda a^{p}+(1-\lambda)b^{p}}{\lambda (a^{p}-b^{p})}\left[\mathbb{B}\left(\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}};1, x\frac{p-1}{p}+1\right)\right.\right. \end{equation*}$

$\begin{equation*} \left.\left.\left.-\mathbb{B}\left(\frac{\lambda (a^{p}-b^{p})}{2(\lambda a^{p} +(1-\lambda)b^{p})};1, x\frac{p-1}{p}+1\right)\right]\right)^{\frac{1}{x}}\right\} \end{equation*}$

$\begin{equation} \times\left\{\frac{2^{s+1}-\lambda}{2^{s+1}(s+1)}|f'(b)|^{y}+\frac{ \lambda}{2^{s+1}(s+1)}|f'(a)|^{y}\right\}^{\frac{1}{y}}. \end{equation}$

(3.11)

Therefore, the desired inequality (3.9) follows from (3.5) and (3.10) together with (3.11).

Corollary 3.2. Let $I\subseteq\mathbb{R}^{+}$ be an interval and $I^{\circ}$ be the interior of $I$ , $a, b\in I^{\circ}$ with $a < b$ , $p < 0$ , $u, v > 0$ , $s\in(0, 1]$ , $g(\xi) = \sqrt[p]{\xi}$ for $\xi > 0$ , $y = \frac{x}{x-1} > 1$ , and $f:I \rightarrow\mathbb{R}$ be a differentiable function on $I^{\circ}$ such that $|f'|^{y}$ is $(s, p)$ -convex. Then the inequality

$\begin{equation*} \left|uf(a)+(1-v)f(b)+(v-u)f\left(\sqrt[p]{\frac{a^{p}+b^{p}}{2}}\right)-\frac{1}{b^{p}-a^{p}}\int_{a^{p}}^{b^{p}}(f{\circ}g)(x)dx\right| \end{equation*}$

$\begin{equation*} \leq\frac{(b^{p}-a^{p})\left(\sqrt[x]{a^{p-x^{2}(p-1)}}\right)}{p \sqrt[x]{a^{p}-b^{p}}\sqrt[y]{2^{1+s}(s+1)}} \left[u\sqrt[x]{\mathbb{B}\left(\frac{ a^{p}-b^{p}}{ 2a^{p}};1, \frac{xp-x+p}{p}\right)}\right. \end{equation*}$

$\begin{equation*} \times\sqrt[y]{|f'(b)|^{y}+ (2^{s+1}-1)|f'(a)|^{y}} \end{equation*}$

$\begin{equation*} +v\sqrt[x]{\mathbb{B}\left(\frac{ a^{p}-b^{p}}{ a^{p}};1, \frac{xp-x+p}{p}\right)-\mathbb{B} \left(\frac{a^{p}-b^{p}}{ 2a^{p}};1, \frac{xp-x+p}{p}\right)} \end{equation*}$

$\begin{equation} \left.\times\sqrt[y]{(2^{s+1}-1)|f'(b)|^{y}+|f'(a)|^{y}}\right] \end{equation}$

(3.12)

holds.

Theorem 3.3. Let $I\subseteq\mathbb{R}^{+}$ be an interval and $I^{\circ}$ be the interior of $I$ , $a, b\in I^{\circ}$ with $a < b$ , $p, \rho, \beta > 0$ , $x\geq 1$ , $u, w\in\mathbb{R}$ , $(s, \lambda)\in(0, 1]\times[0, 1]$ , $g(\xi) = \sqrt[p]{\xi}$ , and $f:I\rightarrow\mathbb{R}$ be a differentiable function on $I^{\circ}$ such that $|f'|$ is $(s, p)$ -convex. Then one has

$\begin{equation*} |\psi(t, a, b;f)|\leq \frac{[\lambda(b^{p}- a^{p})]^{1+\beta}}{\left(\sqrt[x p] {\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \left\{\mathfrak{F}_{\rho, \beta+1}^{\sigma_{3}}\left[|w| \left(\frac{\lambda(b^{p}-a^{p})}{2}\right)^{\rho}\right]\right. \end{equation*}$

$\begin{equation*} +|u|\left\{\frac{\lambda a^{p}+(1-\lambda)b^{p}}{\lambda (a^{p}-b^{p})}\mathbb{B} \left(\frac{\lambda (a^{p}-b^{p})}{2(\lambda a^{p}+(1-\lambda)b^{p})}, 1, \frac{1}{p}\right)\right\}^{\frac{x-1}{x}} \end{equation*}$

$\begin{equation*} \times \sqrt[x]{\frac{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}H_{2, \lambda}(a, b;1, s+2)+|f'( b)|^{x} H_{2, \lambda}(a, b;s+1, s+2)}{s+1}} \end{equation*}$

$\begin{equation*} +|v|\left\{\frac{\lambda a^{p}+(1-\lambda)b^{p}}{\lambda(a^{p}-b^{p})}\left[\mathbb{B}\left(\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}};1, \frac{1}{p}\right)\right.\right. \end{equation*}$

$\begin{equation*} \left.\left.-\mathbb{B}\left(\frac{\lambda (a^{p}-b^{p})}{2(\lambda a^{p}+(1-\lambda)b^{p})};1, \frac{1}{p}\right)\right]\right\}^{\frac{x-1}{x}} \left\{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}\right. \end{equation*}$

$\begin{equation*} \times\frac{H_{1, \lambda}(a, b;1, s+2)- H_{2, \lambda}(a, b;1, s+2)}{s+1} \end{equation*}$

$\begin{equation} \left.\left.+\frac{|f'( b)|^{x}}{s+1}\left[H_{1, \lambda}(a, b;s+1, s+2)- H_{2, \lambda}(a, b;s+1, s+2)\right]\right\}^{\frac{1}{x}}\right\}. \end{equation}$

(3.13)

Proof. It follows from the $(s, p)$ -convexity of $|f'|$ and the power-mean inequality that

$\begin{equation*} \left.\times f'\left(((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p})^{\frac{1}{p}}\right)dt\right| \end{equation*}$

$\begin{equation*} \leq\int_{0}^{\frac{1}{2}}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[|w|\left(\frac{\lambda (b^{p} -a^{p})}{2}\right)^{\rho}t^{\rho}\right]+|u|\right\}[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\left|f'\left(((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p})^{\frac{1}{p}}\right)\right|dt \end{equation*}$

$\begin{equation*} = \int_{0}^{\frac{1}{2}}\left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)}t^{\beta+\rho k}+|u|\right\} \end{equation*}$

$\begin{equation*} \times[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\left|f'\left(\left((1-t)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p}+t b^{p}\right)^{\frac{1}{p}}\right)\right|dt \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k} }{\Gamma(\rho k+\beta+1)} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}t^{\beta+\rho k} \left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}t^{\beta+\rho k} \left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} \left.\times \left|f'\left(\left((1-t)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p} +tb^{p}\right)^{\frac{1}{p}}\right)\right|^{x}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +|u| \left\{\int_{0}^{\frac{1}{2}}\left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p}) +t b^{p}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}\left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}t^{\beta+\rho k} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}t^{\beta+\rho k} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} \left.\times\left[(1-t)^{s}\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}+t^{s}|f'( b)|^{x}\right]dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +\frac{|u|}{\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}\left[1-t\frac{\lambda(a^{p}-b^{p})}{\lambda a^{p} +(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}\left[1-t\frac{\lambda(a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} \left.\times\left[(1-t)^{s}\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}+t^{s}|f'( b)|^{x}\right]dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left\{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}\right. \end{equation*}$

$\begin{equation*} \times\int_{0}^{\frac{1}{2}}t^{\beta+\rho k}(1-t)^{s} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt \end{equation*}$

$\begin{equation*} \left.+|f'( b)|^{x}\int_{0}^{\frac{1}{2}}t^{\beta+\rho k+s} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +\frac{|u|}{\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{0}^{\frac{1}{2}}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \left\{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}\right. \end{equation*}$

$\begin{equation*} \times\int_{0}^{\frac{1}{2}} (1-t)^{s}\left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt \end{equation*}$

$\begin{equation*} \left.+|f'(b)|^{x}\int_{0}^{\frac{1}{2}} t^{s}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda(b^{p}-a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \left\{\mathbb{B}(\beta+\rho k+1, 1)\right. \end{equation*}$

$\begin{equation*} \left.\times{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+1, \beta+\rho k+2;\frac{1}{2}\right], \frac{\lambda(a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'(b)|^{x}\}\mathbb{B}(\beta+\rho k+1, s+1)\right. \end{equation*}$

$\begin{equation*} \times{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+1, s+\beta+\rho k+2;\frac{1}{2}\right], \frac{\lambda(a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)+|f'(b)|^{x} \end{equation*}$

$\begin{equation*} \times{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+s+1, \beta+\rho k+s+2; \frac{1}{2}\right], \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right) \end{equation*}$

$\begin{equation*} \left.\times\mathbb{B}(\beta+\rho k+s+1, 1)\right\}^{\frac{1}{x}} +\frac{|u|}{\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left\{\frac{\lambda a^{p}+(1-\lambda)b^{p}}{\lambda (a^{p}-b^{p})}\mathbb{B} \left(\frac{\lambda(a^{p}-b^{p})}{2(\lambda a^{p}+(1-\lambda)b^{p})}, 1, \frac{1}{p}\right)\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}\mathbb{B}(1, s+1)\right. \end{equation*}$

$\begin{equation*} \times{}_{2}F_{1}\left(\frac{p-1}{p}, \left[1, s+2;\frac{1}{2}\right], \frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right)+|f'( b)|^{x}\mathbb{B}(s+1, 1) \end{equation*}$

$\begin{equation} \left.\times{}_{2}F_{1}\left(\frac{p-1}{p}, \left[s+1, s+2;\frac{1}{2}\right], \frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right)\right\}^{\frac{1}{x}} \end{equation}$

(3.14)

and

$\begin{equation*} \left.\times f'\left(((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p})^{\frac{1}{p}}\right)dt\right| \end{equation*}$

$\begin{equation*} \leq\int_{\frac{1}{2}}^{1}\left\{t^{\beta}\mathfrak{F}_{\rho, \beta+1}^{\sigma}\left[|w| \left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho}t^{\rho}\right]+|v|\right\} [(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\left|f'\left(((1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p})^{\frac{1}{p}}\right)\right|dt \end{equation*}$

$\begin{equation*} = \int_{\frac{1}{2}}^{1}\left\{\sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}- a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)}t^{\beta+\rho k}+|v|\right\} \end{equation*}$

$\begin{equation*} \times[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}]^{\frac{1-p}{p}} \end{equation*}$

$\begin{equation*} \times\left|f'\left(\left((1-t)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p}+t b^{p}\right)^{\frac{1}{p}}\right)\right|dt \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda(b^{p}- a^{p})}{2}\right)^{\rho k} }{\Gamma(\rho k+\beta+1)} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}t^{\beta+\rho k}\left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times \left\{\int_{\frac{1}{2}}^{1}t^{\beta+\rho k} \left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} \left.\times\left|f'\left(\left((1-t)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p} +t b^{p}\right)^{\frac{1}{p}}\right)\right|^{x}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +|v| \left\{\int_{\frac{1}{2}}^{1}\left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}\left[(1-t)(\lambda a^{p}+(1-\lambda)b^{p})+t b^{p}\right]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} \left.\times\left|f'\left(\left((1-t)\left(\sqrt[p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{p} +tb^{p}\right)^{\frac{1}{p}}\right)\right|^{x}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}-a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}t^{\beta+\rho k} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}t^{\beta+\rho k} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} \left.\times\left[(1-t)^{s}\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}+t^{s}|f'( b)|^{x}\right]dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +\frac{|v|}{\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}\left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}\right. \end{equation*}$

$\begin{equation*} \left.\times\left[(1-t)^{s}\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}+t^{s}|f'( b)|^{x}\right]dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda (b^{p}-a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}t^{\beta+\rho k}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}\right. \end{equation*}$

$\begin{equation*} \times\int_{\frac{1}{2}}^{1}t^{\beta+\rho k}(1-t)^{s}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt+|f'( b)|^{x} \end{equation*}$

$\begin{equation*} \left.\times\int_{\frac{1}{2}}^{1}t^{\beta+\rho k+s}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} +\frac{|v|}{\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left\{\int_{\frac{1}{2}}^{1}\left[1-t\frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}\int_{\frac{1}{2}}^{1}(1-t)^{s} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right. \end{equation*}$

$\begin{equation*} \left.+|f'( b)|^{x}\int_{\frac{1}{2}}^{1}t^{s} \left[1-t\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right]^{\frac{1-p}{p}}dt\right\}^{\frac{1}{x}} \end{equation*}$

$\begin{equation*} = \sum\limits_{k = 0}^{\infty}\frac{\sigma(k)|w|^{k}\left(\frac{\lambda(b^{p}-a^{p})}{2}\right)^{\rho k}} {\Gamma(\rho k+\beta+1)\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \left\{\mathbb{B}(\beta+\rho k+1, 1)\right. \end{equation*}$

$\begin{equation*} \times\left[{}_{2}F_{1}\left(\frac{p-1}{p}, \beta+\rho k+1;\beta+\rho k+2, \frac{\lambda(a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right. \end{equation*}$

$\begin{equation*} \left.\left.-{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+1, \beta+\rho k+2;\frac{1}{2}\right], \frac{\lambda(a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]\right\}^{1-\frac{1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'(b)|^{x}\}\mathbb{B}(\beta+\rho k+1, s+1)\right. \end{equation*}$

$\begin{equation*} \times\left[{}_{2}F_{1}\left(\frac{p-1}{p}, \beta+\rho k+1; s+\beta+\rho k+2, \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right. \end{equation*}$

$\begin{equation*} \left.-{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+1, s+\beta+\rho k+2;\frac{1}{2}\right], \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]+|f'(b)|^{x} \end{equation*}$

$\begin{equation*} \times\left[{}_{2}F_{1}\left(\frac{p-1}{p}, \beta+\rho k+s+1;\beta+\rho k+s+2, \frac{\lambda(a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right. \end{equation*}$

$\begin{equation*} \left.-{}_{2}F_{1}\left(\frac{p-1}{p}, \left[\beta+\rho k+s+1, \beta+\rho k+s+2;\frac{1}{2}\right], \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right] \end{equation*}$

$\begin{equation*} \left.\times \mathbb{B}(\beta+\rho k+s+1, 1)\right\}^{\frac{1}{x}} +\frac{|v|}{\left(\sqrt[x p]{\lambda a^{p}+(1-\lambda)b^{p}}\right)^{(x-1)(p-1)}} \end{equation*}$

$\begin{equation*} \times\left[\left\{\frac{\lambda a^{p}+(1-\lambda)b^{p}}{\lambda (a^{p}-b^{p})} \left[\mathbb{B}\left(\frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}};1, \frac{1}{p}\right)\right.\right.\right. \end{equation*}$

$\begin{equation*} \left.\left.-\mathbb{B}\left(\frac{\lambda (a^{p}-b^{p})}{2(\lambda a^{p}+(1-\lambda)b^{p})}; 1, \frac{1}{p}\right)\right]\right\}^{1-\frac{1}{x}} \left\{\{\lambda|f'(a)|^{x}+(1-\lambda)|f'( b)|^{x}\}\right. \end{equation*}$

$\begin{equation*} \times\mathbb{B}(1, s+1)\left[{}_{2}F_{1}\left(\frac{p-1}{p}, 1; s+2, \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right. \end{equation*}$

$\begin{equation*} \left.-{}_{2}F_{1}\left(\frac{p-1}{p}, [1, s+2;\frac{1}{2}], \frac{\lambda (a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]+|f'( b)|^{x}\mathbb{B}(s+1, 1) \end{equation*}$

$\begin{equation*} \times\left[{}_{2}F_{1}\left(\frac{p-1}{p}, s+1; s+2, \frac{\lambda (a^{p}-b^{p})}{\lambda a^{p}+(1-\lambda)b^{p}}\right)\right. \end{equation*}$

$\begin{equation} \left.\left.\left.-{}_{2}F_{1}\left(\frac{p-1}{p}, \left[s+1, s+2;\frac{1}{2}\right], \frac{\lambda(a^{p}-b^{p})} {\lambda a^{p}+(1-\lambda)b^{p}}\right)\right]\right\}^{\frac{1}{x}}\right]. \end{equation}$

(3.15)

Combining the inequalities (3.5), (3.14) and (3.15) gives the desired inequality (3.13).

Corollary 3.3. Let $I\subseteq\mathbb{R}^{+}$ be an interval and $I^{\circ}$ be the interior of $I$ , $a, b\in I^{\circ}$ with $a < b$ , $p < 0$ , $s\in(0, 1]$ , $u, v > 0$ , $g(\xi) = \sqrt[p]{\xi}$ , $y = \frac{x}{x-1} > 1$ and $f:I\rightarrow\mathbb{R}$ be a differentiable function on $I^{\circ}$ such that $|f'|^{y}$ is $(s, p)$ -convex. Then

$\begin{equation*} \left|uf(a)+(1-v)f(b)+(v-u)f\left(\sqrt[p]{\frac{a^{p}+b^{p}}{2}}\right)-\frac{1}{b^{p}-a^{p}}\int_{a^{p}}^{b^{p}}(f{\circ}g)(x)dx\right| \end{equation*}$

$\begin{equation*} \leq\frac{b^{p}-a^{p}}{p\left(\sqrt[x]{a}\right)^{(x-1)(p-1)}}\left\{u\left\{\frac{a^{p}}{a^{p}-b^{p}}\mathbb{B} \left(\frac{a^{p}-b^{p}}{2a^{p}}, 1, \frac{1}{p}\right)\right\}^{\frac{x-1}{x}}\right. \end{equation*}$

$\begin{equation*} \times\sqrt[x]{\frac{|f'(a)|^{x} H_{2, 1}(a, b;1, s+2)+|f'(b)|^{x}H_{2, 1}(a, b;s+1, s+2)}{s+1}} \end{equation*}$

$\begin{equation*} +v\left\{\frac{a^{p}}{a^{p}-b^{p}}\left[\mathbb{B}\left(\frac{a^{p}-b^{p}}{a^{p}};1, \frac{1}{p}\right) -\mathbb{B}\left(\frac{a^{p}-b^{p}}{2a^{p}};1, \frac{1}{p}\right)\right]\right\}^{\frac{x-1}{x}} \end{equation*}$

$\begin{equation*} \times\left\{\frac{|f'(a)|^{x} H_{1, 1}(a, b;1, s+2) H_{2, 1}(a, b;1, s+2)}{s+1}\right. \end{equation*}$

$\begin{equation} \left.\left.+\frac{|f'(b)|^{x}[H_{1, 1}(a, b;s+1, s+2) -H_{2, 1}(a, b;s+1, s+2)]}{s+1}\right\}^{\frac{1}{x}}\right\}. \end{equation}$

(3.16)

Proof. Let $\beta, \lambda, \sigma(0)\rightarrow 1$ and $w = 0$ . Then Corollary 3.3 follows easily from Theorem 3.3.

4. Applications

In this section, we provide some applications on $f$ -divergence measures and probability density functions by using the results obtained in Section 3.

4.1. $f$ -divergence measures

Let $\phi$ be a set, $\mu$ be the $\sigma$ finite measure, $\Omega = \{\chi|\chi: \phi\rightarrow\mathbb{R}, \chi(x) > 0, \int_{\phi}\chi(x) d\mu(x) = 1\}$ be the set of all probability densities on $\mu$ , and $f:(0, \infty)\rightarrow\mathbb{R}$ be a real-valued function. Then the Csis\'{z}ar $f$ -divergence $D_{f} (\chi, \psi)$ is defined by

$\begin{equation} D_{f}(\chi, \psi) = \int_{\phi}\chi(x)f\left[\frac{\psi(x)}{\chi(x)}\right]d\mu(x) \quad (\chi, \psi\in\Omega) \end{equation}$

(4.1)

if $f$ is convex, and the Hermite-Hadamard $(HH)$ divergence $D_{HH}^{f}(\chi, \psi)$ is defined by

$\begin{equation} D_{HH}^{f}(\chi, \psi) = \int_{\phi}\chi(x)\frac{\int_{1}^{\frac{\psi(x)}{\chi(x)}}f(t)dt}{\frac{\psi(x)}{\chi(x)}-1}d\mu(x) \quad (\chi, \psi\in\Omega) \end{equation}$

(4.2)

if $f$ is convex with $f(1) = 0$ . Note that $D_{HH}^{f}(\chi, \psi)\geq 0$ and $D_{HH}^{f}(\chi, \psi) = 0$ if and only if $\chi = \psi$ .

Proposition 4.1. Let $I\subseteq\mathbb{R}^{+}$ be an interval and $I^{\circ}$ be the interior of $I$ , $a, b\in I^{\circ}$ with $a < b$ , $s\in(0, 1]$ and $f:I\rightarrow\mathbb{R}$ be a differentiable function on $I^{\circ}$ such that $|f'|$ is $s$ -convex and $f(1) = 0$ . Then

$\begin{equation*} \left|\frac{1}{6}\left[D_{f}\left(\chi, \psi\right) +4\int_{\phi}\chi(x)f\left(\frac{\psi(x)+\chi(x)}{2\chi(x)}\right)d\mu(x)\right]-D_{HH}^{f}(\chi, \psi)\right| \end{equation*}$

$\begin{equation*} \leq\frac{(b-a)|f'(a)|}{6(s+1)}\int_{\phi}\chi(x)\left[5{}_{2}F_{1}\left(0, 1, s+2, \frac{|\psi(x)-\chi(x)|}{\chi(x)}\right)\right. \end{equation*}$

$\begin{equation*} \left.-4{}_{2}F_{1}\left(0, \left[1, s+2;\frac{1}{2}\right], \frac{|\psi(x)-\chi(x)|}{\chi(x)}\right)\right]d\mu(x) \end{equation*}$

$\begin{equation*} +\frac{(b-a)|f'(b)|}{6(s+1)}\int_{\phi}\chi(x)\left[5{}_{2}F_{1}\left(0, s+1, s+2, \frac{|\psi(x)-\chi(x)|}{\chi(x)}\right)\right. \end{equation*}$

$\begin{equation} \left.-4{}_{2}F_{1}\left(0, \left[s+1, s+2;\frac{1}{2}\right], \frac{|\psi(x)-\chi(x)|}{\chi(x)}\right)\right]d\mu(x). \end{equation}$

(4.3)

Proof. Let $\Phi_{1} = \{x\in\phi:\psi(x) > \chi(x)\}$ , $\Phi_{2} = \{x\in\phi:\psi(x) < \chi(x)\}$ and $\Phi_{3} = \{x\in\phi:\psi(x) = \chi(x)\}$ . Then we clearly see that inequality (4.3) holds if $x\in\Phi_{3}$ .

For the case of $x\in\Phi_{1}$ , taking $a, p\rightarrow1$ , $b\rightarrow\frac{\psi(x)}{\chi(x)}$ , $u\rightarrow\frac{1}{6}$ and $v\rightarrow\frac{5}{6}$ in Corollary 3.1, multiplying both sides of the obtained result by $\chi(x)$ and integrating over $\Phi_{1}$ lead to the conclusion that

$\begin{equation*} \left|\frac{1}{6}\left[4\int_{\Phi_{1}}\chi(x)f\left(\frac{\psi(x)+\chi(x)}{2\chi(x)}\right)d\mu(x) +\int_{\Phi_{1}}\chi(x)f\left(\frac{\psi(x)}{\chi(x)}\right)d\mu(x)\right]\right. \end{equation*}$

$\begin{equation*} \left.-\int_{\Phi_{1}}\chi(x)\frac{\int_{1}^{\frac{\psi(x)}{\chi(x)}}f(t)dt}{\frac{\psi(x)}{\chi(x)}-1}d\mu(x)\right|\leq\frac{(b-a)|f'(a)|}{6(s+1)} \int_{\Phi_{1}}\chi(x)\left[5{}_{2}F_{1}\left(0, 1, s+2, \frac{\chi(x)-\psi(x)}{\chi(x)}\right)\right. \end{equation*}$

$\begin{equation*} \left.-4{}_{2}F_{1}\left(0, \left[1, s+2;\frac{1}{2}\right], \frac{\chi(x)-\psi(x)}{\chi(x)}\right)\right]d\mu(x) \end{equation*}$

$\begin{equation*} +\frac{(b-a)|f'(b)|}{6(s+1)}\int_{\Phi_{1}}\chi(x)\left[5{}_{2}F_{1}\left(0, s+1, s+2, \frac{\chi(x)-\psi(x)}{\chi(x)}\right)\right. \end{equation*}$

$\begin{equation} \left.-4{}_{2}F_{1}\left(0, \left[s+1, s+2;\frac{1}{2}\right], \frac{\chi(x)-\psi(x)}{\chi(x)}\right)\right]d\mu(x). \end{equation}$

(4.4)

Similarly, for the case of $x\in\Phi_{2}$ , taking $b, p\rightarrow1$ , $a\rightarrow\frac{\psi(x)}{\chi(x)}$ , $u\rightarrow\frac{1}{6}$ and $v\rightarrow\frac{5}{6}$ in Corollary 3.1, multiplying both sides to the obtained result by $\chi(x)$ and integrating over $\Phi_{2}$ , we get

$\begin{equation*} \left|\frac{1}{6}\left[4\int_{\Phi_{2}}\chi(x)f\left(\frac{\psi(x)+\chi(x)}{2\chi(x)}\right)d\mu(x) +\int_{\Phi_{2}}\chi(x)f\left(\frac{\psi(x)}{\chi(x)}\right)d\mu(x)\right]\right. \end{equation*}$

$\begin{equation*} \left.-\int_{\Phi_{2}}\chi(x)\frac{\int_{1}^{\frac{\psi(x)}{\chi(x)}}f(t)dt}{\frac{\psi(x)}{\chi(x)}-1}d\mu(x)\right| \leq\frac{(b-a)|f'(a)|}{6(s+1)}\int_{\Phi_{2}}\chi(x)\left[5{}_{2}F_{1}\left(0, 1, s+2, \frac{\psi(x)-\chi(x)}{\chi(x)}\right)\right. \end{equation*}$

$\begin{equation*} \left.-4{}_{2}F_{1}\left(0, \left[1, s+2;\frac{1}{2}\right], \frac{\psi(x)-\chi(x)}{\chi(x)}\right)\right]d\mu(x) \end{equation*}$

$\begin{equation*} +\frac{(b-a)|f'(b)|}{6(s+1)}\int_{\Phi_{2}}\chi(x)\left[5{}_{2}F_{1}\left(0, s+1, s+2, \frac{\psi(x)-\chi(x)}{\chi(x)}\right)\right. \end{equation*}$

$\begin{equation} \left.-4{}_{2}F_{1}\left(0, \left[s+1, s+2;\frac{1}{2}\right], \frac{\psi(x)-\chi(x)}{\chi(x)}\right)\right]d\mu(x). \end{equation}$

(4.5)

Therefore, the desired inequality (4.3) can be derived by adding inequalities (4.4) and (4.5) to together with the triangular inequality.

4.2. Probability density functions

Let $a, b\in\mathbb{R}$ with $a < b$ , $g:[a, b]\rightarrow[0, 1]$ be the probability density function of a continuous random variable $X$ with the cumulative distribution function $F$ given by

$\begin{equation} F(x) = P(X\leq x) = \int_{a}^{x}g(t)dt, \quad E(X) = \int_{a}^{b}tdF(t) = b-\int_{a}^{b}F(t)dt. \end{equation}$

(4.6)

Then from Corollary 3.1 we clearly see that

$\begin{equation*} \left|\frac{1}{6}\left[4P\left(X\leq\frac{a+b}{2}\right)+1\right]-\frac{1}{b-a}(b-E(X))\right| \end{equation*}$

$\begin{equation*} \leq\frac{b-a}{s+1}\left\{|g(a)|\frac{5{}_{2}F_{1}\left(0, 1, s+2, \frac{a-b}{a}\right) -4{}_{2}F_{1}\left(0, \left[1, s+2;\frac{1}{2}\right], \frac{a-b}{a}\right)}{6}\right. \end{equation*}$

$\begin{equation*} \left.+|g(b)|\frac{5.{}_{2}F_{1}\left(0, s+1, s+2, \frac{a-b}{a}\right) -4{}_{2}F_{1}\left(0, \left[s+1, s+2;\frac{1}{2}\right], \frac{a-b}{a}\right)}{6}\right\} \end{equation*}$

if $p\rightarrow1$ , $u\rightarrow\frac{1}{6}$ and $v\rightarrow\frac{5}{6}$ .

5. Conclusions

We have established some new estimates for the generalized Simpson's quadrature rule via the Raina fractional integrals by use of a Simpson-type generalized identity with multi-parameters, and discovered several inequalities for the $f$ -divergence measures and probability density functions. Our obtained results are the improvements and generalizations of some previous known results, our ideas and approach may lead to a lot of follow-up research.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11971142, 11701176, 11626101, 11601485).

Conflict of interest

The authors declare no conflict of interest.

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AIMS Mathematics

Some generalized fractional integral Simpson’s type inequalities with applications

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Abstract

1. Introduction

2. Preliminaries and assumptions

3. Main results

4. Applications

4.1. $f$ -divergence measures

4.2. Probability density functions

5. Conclusions

Acknowledgements

Conflict of interest

References

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AIMS Mathematics

Some generalized fractional integral Simpson’s type inequalities with applications

Related Papers:

Abstract

1. Introduction

2. Preliminaries and assumptions

3. Main results

4. Applications

4.1. f f -divergence measures

4.2. Probability density functions

5. Conclusions

Acknowledgements

Conflict of interest

References

This article has been cited by:

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4.1. $f$ -divergence measures