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Ostrowski-type inequalities for n-polynomial \(\mathscr{P}\)-convex function for k-fractional Hilfer–Katugampola derivative

Abstract

In this article, we develop a novel framework to study a new class of convex functions known as n-polynomial \(\mathscr{P} \)-convex functions. The purpose of this article is to establish a new generalization of Ostrowski-type integral inequalities by using a generalized k-fractional Hilfer–Katugampola derivative. We employ this technique by using the Hölder and power-mean integral inequalities. We present analogs of the Ostrowski-type integrals inequalities connected with the n-polynomial \(\mathscr{P}\)-convex function. Some new exceptional cases from the main results are obtained, and some known results are recaptured. In the end, an application to special means is given as well. The article seeks to create an exciting combination of a convex function and special functions in fractional calculus. It is supposed that this investigation will provide new directions in fractional calculus.

Introduction

In the last few decades, fractional calculus has been viably utilized in models over an enormous assortment of designing and applied science processes and structures. Fractional calculus in, for example, fluid mechanics, including exothermal compound responses or autocatalytic responses, characterizes the broad applications for explicit issues. It was created as a productive strategy for comprehension and demonstrating different issues in material science and applied mathematics. Fragmentary necessary conditions incorporate a derivative of any unpredictable or real requirement, which will likewise be viewed as differing conditions of an overall sort. Many explorations are considered having been proposed to upgrade demonstrating the accuracy while indicating the diffusion process, displaying various types of viscoelastic damping, unequivocally leading to the reliance on power-law frequencies, and modeling fragmentary Maxwell liquid streaming (see [3, 5, 20]).

Integral inequalities are among the best-known techniques relevant for numerous practical points: optimization, design, and innovation. In many fields of science and technology, integral inequalities in fractional strategies are dramatically more common. There has been a focus on the massive degree of the advantages of integral inequalities in taking a derivative and doing integration via convexity [1316].

Presenting the idea of n-polynomial \(\mathscr{P} \)-convex functions and characterizing the Ostrowski-type inequalities for n-polynomial \(\mathscr{P} \)-convex functions are the fundamental subject of this paper. In the deterministic case, the vast majority of the results introduced are the refinements in the general writing of the current outcomes for new and classical convex functions.

This article is organized as follows: in Sect. 2, some basic and essential definitions and lemmas are recalled. In Sect. 3, for n-polynomial \(\mathscr{P} \)-convex functions, we proved an inequality of the Ostwoski type and related results. In Sect. 4, we give our concluding remarks.

Preliminaries

Firstly, we include some mandatory definitions and mathematical preliminaries of the fractional operators of calculus, which are further used in this article.

Definition 2.1

(See [9])

Let \([{a_{1}},{a_{2}}] \) be a finite or infinite interval on the real axis \(\mathbb{R}=(-\infty ,\infty )\). By \({M_{q}} = ( {{a_{1}},{a_{2}}} ) \), we denote the set of the complex-valued Lebesgue measurable function ψ on \([{a_{1}},{a_{2}}] \).

$$\begin{aligned} {M_{q}} ( {{a_{1}},{a_{2}}} ) = \biggl\{ {\psi : \Vert {{\psi _{q}}} \Vert = \sqrt[q]{{ \int {{{ \bigl\vert {\psi ( z )} \bigr\vert }^{q}}} \,dz}} < + \infty } \biggr\} , \quad 1 \le q < \infty . \end{aligned}$$

In the case if \(q = 1\), we have \({M_{q}} ( {{a_{1}},{a_{2}}} ) = M ( {{a_{1}},{a_{2}}} )\).

Definition 2.2

(See [4])

Diaz et al. defined the k-Gamma function as

$$\begin{aligned} {\Gamma _{k} }(z) = \int _{0}^{\infty }{t^{z - 1}} {e^{ - \frac{{{t^{k} }}}{k }}} \, \mathrm{d}t. \end{aligned}$$
(2.1)

Here \(z,k > 0 \). We have \({\Gamma _{k} } ( {z + k } ) = z{\Gamma _{k} } ( z )\) and \(\Gamma _{k} (z)={k}^{{\frac{z}{k}}-1}\Gamma ( {\frac{z}{k}} ) \).

Definition 2.3

(See [17])

Sarikaya et al. presented the left and right generalized k-fractional integral of order ω with \(m - 1 < \omega \le m\), \(m \in \mathbb{N}\), \(\rho > 0\), \(k >0\), \(\omega >0 \). We have

$$\begin{aligned} & \bigl( {{}_{k} ^{\rho }\Im _{a_{1}}^{\omega }\psi } \bigr) ( z ) = \frac{{{\rho ^{1 - \frac{\omega }{k }}}}}{{k {\Gamma _{k} } ( \omega )}} \int _{a_{1}}^{z} {{{ \bigl( {{z^{\rho }} - {y^{\rho }}} \bigr)}^{ \frac{\omega }{k } - 1}}} {y^{\rho - 1}}\psi ( y )\,dy, \quad z>{a_{1}}, \end{aligned}$$
(2.2)
$$\begin{aligned} & \bigl( {{}_{k} ^{\rho }\Im _{a_{2}}^{\omega }\psi } \bigr) ( z ) = \frac{{{\rho ^{1 - \frac{\omega }{k }}}}}{{k {\Gamma _{k} } ( \omega )}} \int _{z}^{a_{2}} {{{ \bigl( {{y^{\rho }} - {z^{\rho }}} \bigr)}^{ \frac{\omega }{k } - 1}}} {y^{\rho - 1}}\psi ( y )\,dy, \quad z< {a_{2}}. \end{aligned}$$
(2.3)

Definition 2.4

(See [2])

Nisar et al. presented the left and right generalized k-fractional derivative of order ω can be written as in terms of the integral defined in definition (2.3),

$$\begin{aligned} & {}_{k} ^{\rho }{\mathscr{D}}_{a_{1}}^{\gamma }\psi ( z ) = { \biggl( {{z^{1 - \rho }}\frac{d}{{dz}}} \biggr)^{m}} \bigl( {{k ^{m}}.{}_{k} ^{\rho }\Im _{a_{1}}^{k m - \omega } \psi } \bigr) ( z ),\quad z>{a_{1}}, \end{aligned}$$
(2.4)
$$\begin{aligned} & {}_{k} ^{\rho }{\mathscr{D}}_{a_{2}}^{\gamma }\psi ( z ) = { \biggl( {{z^{1 - \rho }}\frac{d}{{dz}}} \biggr)^{m}} \bigl( {{k ^{m}}.{}_{k} ^{\rho }\Im _{a_{2}}^{k m - \omega }\psi } \bigr) ( z ), \quad z< {a_{2}}. \end{aligned}$$
(2.5)

Definition 2.5

(See [10])

Let \(m-1<\omega \leq m \), \(0\leq \theta \leq 1 \), \(m \in \mathbb{N} \), \(\rho >0 \), \(k>0 \) and \(\psi \in {M_{q}} ( {a,b} )\), the generalized k-fractional Hilfer–Katugampola derivative (left sided and right sided) is defined as

$$\begin{aligned} & \bigl( {_{k} ^{\rho }{\mathscr{D}}_{a_{1}}^{\omega ,\theta } \psi } \bigr) ( z ) = \biggl( {_{k} ^{\rho }\Im _{a_{1}}^{ \theta ( {k m - \omega } )}{{ \biggl( {{z^{1 - \rho }} \frac{d}{{dz}}} \biggr)}^{m}} \bigl( {{k ^{m}}.{}_{k} ^{\rho }\Im _{a_{1}}^{ ( {1 - \theta } ) ( {k m - \omega } )}\psi } \bigr)} \biggr) ( z ), \end{aligned}$$
(2.6)
$$\begin{aligned} & \bigl( {_{k} ^{\rho }{\mathscr{D}}_{a_{2}}^{\omega ,\theta } \psi } \bigr) ( z ) = \biggl( {_{k} ^{\rho }\Im _{a_{2}}^{ \theta ( {k m - \omega } )}{{ \biggl( {{z^{1 - \rho }} \frac{d}{{dz}}} \biggr)}^{m}} \bigl( {{k ^{m}}.{}_{\rho }^{k} \Im _{a_{2}}^{ ( {1 - \theta } ) ( {k m - \omega } )}\psi } \bigr)} \biggr) ( z ), \end{aligned}$$
(2.7)

where is the integral defined in definition (2.3).

Lemma 2.1

(See [11])

Let \(m-1<\omega \leq m \), \(0\leq \theta \leq 1 \), \(m \in \mathbb{N} \), \(\rho >0 \), \(k>0 \) and \(\psi \in {M_{q}} ( {a,b} )\), then

$$\begin{aligned} {}_{k} ^{\rho }{\mathscr{D}}_{a_{1}}^{\omega ,\theta } \psi ( z ) &= \biggl( {{}_{k} ^{\rho }\Im _{a_{1}}^{\theta ( {k m - \omega } )}{{ \biggl( {{z^{1 - \rho }}\frac{d}{{dz}}} \biggr)}^{m}} \bigl( {{k ^{m}}.{}_{k} ^{\rho }\Im _{a_{1}}^{ ( {1 - \theta } ) ( {k m - \omega } )} \psi } \bigr)} \biggr) ( z ) \\ &= \biggl( {{}_{k} ^{\rho }\Im _{a_{1}}^{\theta ( {k m - \omega } )}{{ \biggl( {{z^{1 - \rho }}\frac{d}{{dz}}} \biggr)}^{m}} \bigl( {{k ^{m}}.{}_{k} ^{\rho }\Im _{a_{1}}^{k m - \omega - \theta ( {k m - \omega } )}} \bigr)} \biggr)\psi ( z ) \\ &= \biggl( {{}_{k} ^{\rho }\Im _{a_{1}}^{\theta ( {k m - \omega } )}{{ \biggl( {{z^{1 - \rho }}\frac{d}{{dz}}} \biggr)}^{m}} \bigl( {{k ^{m}}.{}_{k} ^{\rho }\Im _{a_{1}}^{k m - \{ { \omega + \theta ( {k m - \omega } )} \} } \psi } \bigr)} \biggr) ( z ) \\ &= \bigl( {{}_{k} ^{\rho }\Im _{a_{1}}^{\theta ( {k m - \omega } )}{}_{k} ^{\rho }D_{a_{1}}^{\omega + \theta ( {k m - \omega } )}\psi } \bigr) ( z ) \quad \bigl( \textit{by using Eq. (2.4)} \bigr) \\ &= \bigl( {{}_{k} ^{\rho }\Im _{a_{1}}^{\gamma - \omega }{}_{k} ^{\rho }D_{a_{1}}^{\gamma }\psi } \bigr) ( z ) \\ &= \bigl( {{}_{k} ^{\rho }\Im _{a_{1}}^{\gamma - \omega }{ \psi ^{ ( \gamma )}}} \bigr) ( z ) \\ &= \frac{{{\rho ^{1 - \frac{{\gamma - \omega }}{k }}}}}{{k {\Gamma _{k} } ( {\gamma - \omega } )}} \int _{a_{1}}^{z} {{{ \bigl( {{z^{\rho }} - {y^{\rho }}} \bigr)}^{ \frac{{\gamma - \omega }}{k } - 1}}} {y^{\rho - 1}} {\psi ^{ ( \gamma )}} ( y )\,dy \quad \bigl(\textit{by using Eq. (2.2)} \bigr), \end{aligned}$$

where \(\gamma = \omega + \theta ( {k m - \omega } )\) and \(\omega >0 \) and \({\psi ^{ ( \gamma )}}\) is the derivative of ψ defined in (2.4).

So the above defined generalized k-fractional Hilfer–Katugampola derivative can be written as

$$\begin{aligned} & \bigl( {_{k} ^{\rho }{\mathscr{D}}_{a_{1}}^{\omega ,\gamma } \psi } \bigr) ( z ) = \frac{{{\rho ^{1 - \frac{{\gamma - \omega }}{k }}}}}{{k {\Gamma _{k} } ( {\gamma - \omega } )}} \int _{a_{1}}^{z} {{{ \bigl( {{z^{\rho }} - {y^{\rho }}} \bigr)}^{ \frac{{\gamma - \omega }}{k } - 1}}} {y^{\rho - 1}} {\psi ^{ ( \gamma )}} ( y )\,dy, \quad z>{a_{1}}, \end{aligned}$$
(2.8)
$$\begin{aligned} & \bigl( {_{k} ^{\rho }{\mathscr{D}}_{a_{2}}^{\omega ,\gamma } \psi } \bigr) ( z ) = \frac{{{\rho ^{1 - \frac{{\gamma - \omega }}{k }}}}}{{k {\Gamma _{k} } ( {\gamma - \omega } )}} \int _{z}^{a_{2}} {{{ \bigl( {{y^{\rho }} - {z^{\rho }}} \bigr)}^{ \frac{{\gamma - \omega }}{k } - 1}}} {y^{\rho - 1}} {\psi ^{ ( \gamma )}} ( y )\,dy, \quad z< {a_{2}}. \end{aligned}$$
(2.9)

Some novel definitions and generalized fractional derivative are presented in this section. The Hermite–Hadamard inequality for a convex function \(\varsigma ^{( \gamma )}: I\rightarrow \mathbb{R} \) is

$$\begin{aligned} \varsigma ^{( \gamma )} \biggl(\frac{{a_{1}}+{a_{2}}}{2} \biggr)\leq ({a_{2}}-{a_{1}})^{-1} \int _{a_{1}}^{a_{2}} \varsigma ^{( \gamma )}(z) \,dz \leq \frac{\varsigma ^{( \gamma )}({a_{1}})+\varsigma ^{( \gamma )}({a_{2}})}{2} \end{aligned}$$
(2.10)

with \(\forall {a_{1}}, {a_{2}}\in I\) with \({a_{1}}\neq {a_{2}}\).

Ostrowski [12] established the integral inequality in 1928 for the integral average \(({a_{2}}-{a_{1}})^{-1}\int _{a_{1}}^{a_{2}} \varsigma ^{( \gamma )}(\eta ) \,d\eta \) by the value \(\varsigma ^{( \gamma )}(z) \) at \(z\in [{a_{1}},{a_{2}}]\).

For \(\varsigma ^{( \gamma )}:[{a_{1}}, {a_{2}}]\rightarrow \mathbb{R}\) a differentiable mapping on \(({a_{1}}, {a_{2}})\) such that \(\vert \varsigma ^{( \gamma +1)}(x) \vert \leq \mathcal{M}\), \(\forall z \in ({a_{1}},{a_{2}})\), the inequality

$$\begin{aligned} \biggl\vert \varsigma ^{( \gamma )}(z)-({a_{2}}-{a_{1}})^{-1} \int _{a_{1}}^{a_{2}} \varsigma ^{( \gamma )}(\eta ) d \eta \biggr\vert \leq \mathcal{M}({a_{2}}-{a_{1}}) \biggl[ \frac{1}{4}+ \frac{(z-\frac{{a_{1}}+{a_{2}}}{2})^{2}}{({a_{2}}-{a_{1}}{})^{2}} \biggr] \end{aligned}$$
(2.11)

holds for \(\forall z \in ({a_{1}}, {a_{2}})\). Here the constant \(\frac{1}{4} \) is the least possible value.

Theorem 2.1

(See [7])

For \(\lambda , \vartheta >1\) with \(\lambda +\vartheta =\lambda \vartheta \), and \(\varsigma ^{( \gamma )} _{1}\) and \(\varsigma ^{( \gamma )} _{2}\) two integrable real-valued functions defined on \([{a_{1}}, {a_{2}}]\) such that \(\vert \varsigma ^{( \gamma )} _{1} \vert ^{\lambda }\) and \(\vert \varsigma ^{( \gamma )} _{2} \vert ^{\vartheta }\) are integrable on \([{a_{1}}, {a_{2}}]\), we have

$$ \begin{aligned} & \int _{a_{1}}^{a_{2}} \bigl\vert \varsigma ^{( \gamma )} _{1}(z)\varsigma ^{( \gamma )} _{2}(z) \bigr\vert \,dz \\ & \quad \leq \frac{1}{{a_{2}}-{a_{1}}} \biggl[ \biggl( \int _{a_{1}}^{a_{2}}({a_{2}}-z) \bigl\vert \varsigma ^{( \gamma )} _{1}(z) \bigr\vert ^{\lambda } \,dz \biggr)^{1/ \lambda } \biggl( \int _{a_{1}}^{a_{2}}({a_{2}}-z) \bigl\vert \varsigma ^{( \gamma )} _{2}(z) \bigr\vert ^{\vartheta } \,dz \biggr)^{1/\vartheta } \\ & \qquad {}+ \biggl( \int _{a_{1}}^{a_{2}}(z-{a_{1}}) \bigl\vert \varsigma ^{( \gamma )} _{1}(z) \bigr\vert ^{\lambda } \,dz \biggr)^{1/\lambda } \biggl( \int _{a_{1}}^{a_{2}}(z-{a_{1}}) \bigl\vert \varsigma ^{( \gamma )} _{2}(z) \bigr\vert ^{\vartheta } \,dz \biggr)^{1/\vartheta } \biggr]. \end{aligned} $$
(2.12)

Theorem 2.2

(See [8])

For \(\lambda , \vartheta >1\) with \(\lambda +\vartheta =\lambda \vartheta \), and \(\varsigma ^{( \gamma )} _{1}\) and \(\varsigma ^{( \gamma )} _{2}\) two integrable real-valued functions defined on \([{a_{1}}, {a_{2}}]\) such that \(\vert \varsigma ^{( \gamma )} _{1} \vert ^{\lambda }\) and \(\vert \varsigma ^{( \gamma )} _{2} \vert ^{\vartheta }\) are integrable on \([{a_{1}}, {a_{2}}]\)

$$\begin{aligned} \begin{aligned} & \int _{a_{1}}^{a_{2}} \bigl\vert \varsigma ^{( \gamma )} _{1}(z)\varsigma ^{( \gamma )} _{2}(z) \bigr\vert \,dz \\ & \quad \leq \frac{1}{{a_{2}}-{a_{1}}} \biggl[ \biggl( \int _{a_{1}}^{a_{2}}({a_{2}}-z) \bigl\vert \varsigma ^{( \gamma )} _{1}(z) \bigr\vert ^{\lambda } \,dz \biggr)^{1-1/ \lambda } \biggl( \int _{a_{1}}^{a_{2}}({a_{2}}-z) \bigl\vert \varsigma ^{( \gamma )} _{1}(z) \bigr\vert \bigl\vert \varsigma ^{( \gamma )} _{2}(z) \bigr\vert ^{\vartheta } \,dz \biggr)^{1/ \vartheta } \\ & \qquad {} + \biggl( \int _{a_{1}}^{a_{2}}(z-{a_{1}}) \bigl\vert \varsigma ^{( \gamma )} _{1}(z) \bigr\vert ^{\lambda } \,dz \biggr)^{1-1/\lambda } \\ & \qquad {} \times \biggl( \int _{a_{1}}^{a_{2}}(z-{a_{1}}) \bigl\vert \varsigma ^{( \gamma )} _{1}(z) \bigr\vert \bigl\vert \varsigma ^{( \gamma )} _{2}(z) \bigr\vert ^{ \vartheta } \,dz \biggr)^{1/\vartheta } \biggr]. \end{aligned} \end{aligned}$$
(2.13)

Definition 2.6

(See [21])

For \(\mathscr{P}>0\) and \(\Omega \subseteq \mathbb{R}\) an interval, Ω is said to be \(\mathscr{P}\)-convex if with \(\forall {a_{1}},{a_{2}}\in \Omega \) and \(\eta \in [0,1]\)

$$\begin{aligned} \bigl(\eta {a_{1}}^{\mathscr{P}}+(1-\eta ) {a_{2}}^{ \mathscr{P}} \bigr)^{{1}/{ \mathscr{P}}}\in \Omega \end{aligned}$$
(2.14)

Definition 2.7

(See [21])

For \(\mathscr{P}>0\) and \(\Omega \subseteq \mathbb{R}\) a \(\mathscr{P}\)-convex interval, the real-valued function \(\varsigma ^{(\gamma )} :\Omega \rightarrow \mathbb{R}\) is said to be \(\mathscr{P}\)-convex if the following inequality holds with \(\forall {a_{1}},{a_{2}}\in \Omega \) and \(\eta \in [0,1]\):

$$\begin{aligned} \varsigma ^{( \gamma )} \bigl( \bigl[\eta {a_{1}}^{ \mathscr{P}}+(1- \eta ) {a_{2}}^{\mathscr{P}} \bigr]^{{1}/{ \mathscr{P}}} \bigr)\leq \eta \varsigma ^{( \gamma )} ({a_{1}})+(1- \eta )\varsigma ^{( \gamma )} ({a_{2}}). \end{aligned}$$
(2.15)

Definition 2.8

(See [6])

For \(\Omega \subseteq \mathbb{R}\) an interval, a real-valued function \(\varsigma ^{( \gamma )} :\Omega \rightarrow \mathbb{R}\) is said to be harmonically convex if the following inequality holds for \(\forall {a_{1}},{a_{2}}\in \Omega \) and \(\eta \in [0,1]\):

$$\begin{aligned} \varsigma ^{( \gamma )} \biggl( \frac{{a_{1}}{a_{2}}}{\eta {a_{1}}+(1-\eta ){a_{2}}} \biggr)\leq \eta \varsigma ^{( \gamma )} ({a_{2}})+(1-\eta ) \varsigma ^{( \gamma )} ({a_{1}}). \end{aligned}$$
(2.16)

Definition 2.9

(See [19])

For \(n\in \mathbb{N}\), the non-negative function \(\varsigma ^{( \gamma )} :\Omega \rightarrow [0, \infty ) \) is said to be a n-polynomial convex function if the following inequality holds for \(\forall {a_{1}},{a_{2}}\in \Omega \) and \(\eta \in [0,1]\):

$$\begin{aligned} \varsigma ^{( \gamma )} \bigl(\eta {a_{1}}+(1-\eta ){a_{2}} \bigr)\leq \frac{1}{n}\sum _{ \theta =1}^{n} \bigl[1-(1-\eta )^{ \theta } \bigr] \varsigma ^{( \gamma )} ({a_{1}}) + \frac{1}{n} \sum _{\theta =1}^{n} \bigl[ \bigl(1-\eta ^{\theta } \bigr) \bigr]\varsigma ^{( \gamma )} ({a_{2}}). \end{aligned}$$
(2.17)

Definition 2.10

(See [1])

For \(n\in \mathbb{N}\), the non-negative function \(\varsigma ^{( \gamma )}:\Omega \rightarrow [0, \infty )\) is said to be a n-polynomial harmonically convex function if the following inequality holds with \(\forall {a_{1}},{a_{2}}\in \Omega \) and \(\eta \in [0,1]\):

$$\begin{aligned} \begin{gathered} \varsigma ^{( \gamma )} \biggl( \frac{{a_{1}}{a_{2}}}{\eta {a_{1}}+(1-\eta ){a_{2}}} \biggr)\\\quad \leq \frac{1}{n}\sum_{\vartheta =1}^{n} \bigl[1-(1-\eta )^{ \vartheta } \bigr] \varsigma ^{( \gamma )}({a_{2}}) + \frac{1}{n}\sum_{\vartheta =1}^{n} \bigl[ \bigl(1-\eta ^{ \vartheta } \bigr) \bigr]\varsigma ^{( \gamma )}({a_{1}}).\end{gathered} \end{aligned}$$
(2.18)

For \(n=2 \), we have

$$\begin{aligned} \varsigma ^{( \gamma )} \biggl( \frac{{a_{1}}{a_{2}}}{\eta {a_{1}}+(1-\eta ){a_{2}}} \biggr)\leq \frac{3\eta -\eta ^{2}}{2}\varsigma ^{( \gamma )}({a_{2}}) + \frac{2-\eta -\eta ^{2}}{2} \varsigma ^{( \gamma )}({a_{1}}). \end{aligned}$$
(2.19)

Definition 2.11

(See [17])

Let \(n\in \mathbb{N}\), \(\mathscr{P}>0\) and \(\Omega \subseteq \mathbb{R}\) be a \(\mathscr{P}\)-convex interval. Then the non-negative real-valued function \(\varsigma ^{( \gamma )}:\Omega \rightarrow [0, \infty )\) is said to be a n-polynomial \(\mathscr{P}\)-convex function if the following inequality holds with \(\forall {a_{1}},{a_{2}}\in \Omega \) and \(\eta \in [0,1]\):

$$\begin{aligned} \begin{gathered} \varsigma ^{( \gamma )} \bigl( \bigl[\eta {a_{1}}^{ \mathscr{P}}+(1- \eta ) {a_{2}}^{\mathscr{P}} \bigr]^{{1}/{ \mathscr{P}}} \bigr)\\\quad \leq \frac{1}{n}\sum_{\vartheta =1}^{n} \bigl[1-(1- \eta )^{\vartheta } \bigr]\varsigma ^{( \gamma )}({a_{1}}) + \frac{1}{n}\sum_{ \vartheta =1}^{n} \bigl[ \bigl(1-\eta ^{\vartheta } \bigr) \bigr]\varsigma ^{( \gamma )}({a_{2}}).\end{gathered} \end{aligned}$$
(2.20)

Remark 2.1

From Definition 2.11 we conclude:

(i) If \(\mathscr{P}=-1\), then Definition 2.11 becomes Definition 2.5 for an n-polynomial harmonically convex function.

(ii) If \(\mathscr{P}=1\), then Definition 2.11 reduces to Definition (2.4) for an n-polynomial convex function.

Definition 2.12

The beta function \(\mathbb{B}\) and Gaussian hypergeometric function \(\mathscr{F}_{1}\) are defined by

$$\begin{aligned} \mathbb{B}(z_{1},z_{2})= \frac{\Gamma (z_{1})\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})} = \int _{0}^{1} \eta ^{z_{1}-1}(1-\eta )^{z_{2}-1} \,d\eta \quad (z_{1},z_{2}>0) \end{aligned}$$
(2.21)

and

$$ \begin{aligned} & \mathscr{F}_{1}(z_{1},z_{2};z_{3},z) = \frac{1}{\mathbb{B}(z_{2},z_{3}-z_{2})} \int _{0}^{1}\eta ^{z_{2}-1} (1- \eta )^{z_{3}-z_{2}-1}(1-z\eta )^{-z_{1}} \,d\eta \\ & \quad \bigl({}z_{3}>z_{2}>0, \vert z \vert < 1 \bigr), \end{aligned} $$
(2.22)

respectively, where \(\Gamma (z)=\int _{0}^{\infty }e^{-\eta }\eta ^{z-1} \,d\eta \) is the Euler Gamma function.

Ostrowski-type inequalities for n-polynomial \(\mathscr{P} \)-convex function

In this section the Ostrowski inequality is proved for an n-polynomial \(\mathscr{P} \)-convex function via a generalized k-fractional Hilfer–Katugampola derivative.

Lemma 3.1

(See [18])

For a differentiable function \(\mu >0\), \(\mathscr{P}\in \mathbb{R}\setminus \{0\}\) and \(\psi ^{( \mu )} :\Omega \rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\) and \(\psi ^{( \mu +1)}\in {M}([{a_{1}},{a_{2}}])\), the following inequality holds:

$$\begin{aligned} \begin{aligned} & \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} -\frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu }\psi \bigr) (z) \bigr] \\ & \quad =- \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{ \zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{ \zeta {a_{2}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta . \end{aligned} \end{aligned}$$
(3.1)

Theorem 3.1

For a differentiable function \(n\in \mathbb{N}\), \(\mu >0\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert \) a n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \\ &\quad\quad{} \times \frac{1}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\mu +2\theta +1}{(\mu +1)(\mu +\theta +1)}-\mathbb{B}(\mu -1, \theta -1) \biggr], \end{aligned} $$
(3.2)

and the following inequality holds with \(\forall z\in ({a_{1}},{a_{2}}) \) and \(\mathscr{P}\in (- \infty , 0)\cup (0, 1)\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{2}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }} \biggl[ \frac{(z^{\mathscr{P}} -{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \\ &\quad\quad{} \times \frac{1}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\mu +2\theta +1}{(\mu +1)(\mu +\theta +1)} -\mathbb{B}(\mu -1, \theta -1) \biggr]. \end{aligned} $$
(3.3)

Proof

We use Lemma 3.1 to prove the inequality (3.2) for the n-polynomial \(\mathscr{P} \)-convexity of \(\vert \psi ^{( \mu +1)} \vert \) to yield

$$\begin{aligned} \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{ \zeta {a_{2}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \\ & \qquad {}\times \Biggl[\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-(1- \zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-\zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert \Biggr] \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \\ & \qquad {}\times \Biggl[ \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-(1- \zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{2}}) \bigr\vert + \frac{1}{n}\sum_{ \theta =1}^{n} \bigl[1-\zeta ^{ \theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert \Biggr] \,d\zeta . \end{aligned} \end{aligned}$$
(3.4)

As \(\mathscr{P}\in (1,\infty )\), we can infer that

$$ \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}}\leq \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}}\leq {a_{1}}^{1- \mathscr{P}}. $$
(3.5)

We proceed by simplifying

$$ \begin{aligned} & \int _{0}^{1}\zeta ^{\mu } \Biggl[ \frac{1}{n}\sum_{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] +\frac{1}{n}\sum _{ \theta =1}^{n} \bigl[1-\zeta ^{\theta } \bigr] \Biggr] \,d\zeta \\ & \quad =\frac{1}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\mu +2\theta +1}{(\mu +1)(\mu +\theta +1)}-\mathbb{B}(\mu -1, \theta -1) \biggr]. \end{aligned} $$
(3.6)

The first inequality of Theorem 3.1 is proved. Now for the second part, we let \(\mathscr{P}\in (-\infty , 0)\cup (0,1)\) to yield

$$ \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}}\leq \bigl(\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}}\leq {a_{2}}^{1- \mathscr{P}}. $$
(3.7)

The above inequality completes the proof of the second part of Theorem 3.1. □

Theorem 3.2

For a differentiable function \(n \in \mathbb{N}\), \(\lambda , \vartheta >1 \) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \(a_{1}, a_{2}\in \Omega \) with \(a_{1}< a_{2}\), and \(\psi ^{( \mu )} :\Omega \subset (0,\infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([a_{1},a_{2}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) an n-polynomial convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [a_{1},a_{2}]\), the following inequality holds for all \(z\in (a_{1},a_{2})\) and \(\mathscr{P}\in (1, \infty )\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}} \mathscr{Q}}{{\mathscr{P}}^{1+\mu }(1+\lambda \mu )^{1/\lambda }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1} +({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \Biggl( \frac{1}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{1/\vartheta }, \end{aligned} $$
(3.8)

and the following inequality holds with \(\forall z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (-\infty , 0)\cup (0, 1)\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{2}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }(1+\lambda \mu )^{1/\lambda }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \Biggl( \frac{1}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{1/\vartheta }. \end{aligned} $$
(3.9)

Proof

We use Lemma 3.1, (3.5) and the Hölder inequality to prove the first inequality of Theorem 3.3,

$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} -\frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {} + \frac{{a_{2}}^{1-\mathscr{P}}({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \biggl( \int _{0}^{1}\zeta ^{\lambda \mu } \,d\zeta \biggr)^{1/ \lambda } \biggl( \int _{0}^{1} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/ \vartheta } \\ & \qquad {}+ \frac{{a_{2}}^{1-\mathscr{P}}({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \biggl( \int _{0}^{1} \zeta ^{\lambda \mu } \,d\zeta \biggr)^{1/ \lambda } \biggl( \int _{0}^{1} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/ \vartheta }. \end{aligned}$$
(3.10)

As \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) is n-polynomial \(\mathscr{P} \)-convex and \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q} \forall z \in [{a_{1}}, {a_{2}}]\), we have

$$ \begin{aligned} & \int _{0}^{1} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1} \Biggl[\frac{1}{n}\sum _{\theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta }+\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr] \,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl[2-(1-\zeta )^{\theta }- \zeta ^{\theta } \bigr] d \zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \end{aligned} $$
(3.11)

and

$$ \int _{0}^{1} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \leq \frac{\mathscr{Q}^{\vartheta }}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1}. $$
(3.12)

Since \(\int _{0}^{1}\zeta ^{\lambda \mu } \,d\zeta = \frac{1}{\lambda \mu +1}\). We get the first inequality of Theorem 3.3 by combining all above inequalities. Continuing in the same way the second inequality of Theorem 3.3 can be proved. □

Theorem 3.3

For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0,\infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) an n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }(1+\lambda \mu )^{1/\lambda }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \\ & \qquad {}\times \Biggl(\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{ \theta =1}^{n} \biggl[ \frac{\vartheta \mu +2\theta +1}{(\mu \vartheta +1)(\mu \vartheta +\theta +1)} - \mathbb{B}(\theta +1,\vartheta \mu +1) \biggr] \Biggr)^{1/ \vartheta } \end{aligned} $$
(3.13)

and the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (-\infty , 0) \cup (0,1)\):

$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} -\frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{2}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }(1+\lambda \mu )^{1/\lambda }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \\ & \qquad {} \times \Biggl(\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{ \theta =1}^{n} \biggl[ \frac{\vartheta \mu +2\theta +1}{(\mu \vartheta +1)(\mu \vartheta +\theta +1)} - \mathbb{B}(\theta +1,\vartheta \mu +1) \biggr] \Biggr)^{1/ \vartheta } . \end{aligned}$$
(3.14)

Proof

We use the Lemma 3.1, to prove Theorem 3.3 and the power-mean inequality to yield

$$\begin{aligned} \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {} + \frac{{a_{2}}^{1-\mathscr{P}}({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl( \int _{0}^{1}\zeta ^{ \vartheta \mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {} + \frac{{a_{2}}^{1-\mathscr{P}}({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl( \int _{0}^{1}\zeta ^{\vartheta \mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta }. \end{aligned} \end{aligned}$$
(3.15)

As \(\vert \psi ^{( \mu )} \vert ^{\vartheta }\) is n-polynomial \(\mathscr{P} \)-convex and \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z \in [{a_{1}}, {a_{2}}]\), we obtain

$$\begin{aligned} \begin{aligned} & \int _{0}^{1}\zeta ^{\vartheta \mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}\zeta ^{\vartheta \mu } \Biggl[ \frac{1}{n} \sum_{\theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1 )}({a_{1}}) \bigr\vert ^{\vartheta }+ \frac{1}{n}\sum _{\theta =1}^{n} \bigl[1-\zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr]\,d \zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl[2\zeta ^{\mu \vartheta } -\zeta ^{\mu \vartheta }(1- \zeta )^{ \theta }+\zeta ^{\vartheta \mu } \bigl(1-\zeta ^{\theta } \bigr) \bigr]\,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\vartheta \mu +2\theta +1}{(\mu \vartheta +1) (\mu \vartheta +\theta +1)}- \mathbb{B}(\theta +1,\vartheta \mu +1) \biggr]. \end{aligned} \end{aligned}$$
(3.16)

Similarly,

$$ \begin{aligned} & \int _{0}^{1}\zeta ^{\vartheta \mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\vartheta \mu +2\theta +1}{(\mu \vartheta +1)(\mu \vartheta +\theta +1)}- \mathbb{B}(\theta +1,\vartheta \mu +1) \biggr]. \end{aligned} $$
(3.17)

We arrive at the first inequality of Theorem 3.3 by combining all above inequalities. For the second part continuing in the same fashion, we find the required result. □

Theorem 3.4

For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) be an n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1,\infty )\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \frac{({a_{1}}^{\lambda (1-\mathscr{P})})}{\lambda (\lambda \mu +1)}+ \frac{1}{\vartheta } \Biggl(\frac{\mathscr{Q}}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{\vartheta } \Biggr] \end{aligned} $$
(3.18)

and the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (-\infty , 0) \cup (0,1)\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \frac{({a_{2}}^{\lambda (1-\mathscr{P})})}{\lambda (\lambda \mu +1)} + \frac{1}{\vartheta } \Biggl(\frac{\mathscr{Q}}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{\vartheta } \Biggr]. \end{aligned} $$
(3.19)

Proof

The Young inequality is \(cd\leq \frac{1}{\lambda }c^{\lambda }+\frac{1}{\vartheta }d^{ \lambda }\), \(c,d\geq 0\), \(\lambda ,\vartheta >1\), \(\lambda ^{-1}+ \vartheta ^{-1}=1\). We use Lemma 3.1 to prove the first part of Theorem 3.3, and using the n-polynomial \(\mathscr{P}\)-convexity of \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) we find

$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \int _{0}^{1} \biggl(\frac{1}{\lambda } \bigl\vert \zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert ^{ \lambda }+ \frac{1}{\vartheta } \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \biggr) \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \int _{0}^{1} \biggl(\frac{1}{\lambda } \bigl\vert \zeta ^{\mu } \bigl( \zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert ^{ \lambda }+ \frac{1}{\vartheta } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \biggr) \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl(\frac{\zeta ^{\lambda \mu }}{\lambda } \bigl\vert \bigl( \zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert ^{\lambda } \\ & \qquad {} +\frac{1}{\vartheta } \Biggl\vert \frac{1}{n}\sum _{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1- \zeta ^{ \theta } \bigr] \Biggr\vert \psi ^{( \mu +1)}(z) \vert \vert ^{\vartheta } \Biggr) \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1} \Biggl(\frac{\zeta ^{\lambda \mu }}{\lambda } \bigl\vert \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert ^{ \lambda } \\ & \qquad {} +\frac{1}{\vartheta } \Biggl\vert \frac{1}{n}\sum _{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{2}}) \bigr\vert + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1- \zeta ^{ \theta } \bigr] \Biggr\vert \psi ^{( \mu +1)}(z) \vert \vert ^{\vartheta } \Biggr) \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \frac{({a_{1}}^{\lambda (1-\mathscr{P})})}{\lambda (\lambda \mu +1)}+ \frac{1}{\vartheta } \Biggl(\frac{\mathscr{Q}}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{\vartheta } \Biggr]. \end{aligned}$$
(3.20)

Continuing in the same fashion, we can prove the second part. □

Theorem 3.5

For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) a n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \frac{\lambda {a_{1}}^{1-\mathscr{P}}}{(\mu +1)}+ \frac{\vartheta \mathscr{Q}}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr] \end{aligned} $$
(3.21)

and the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (-\infty , 0) \cup (0,1)\),

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[\frac{\lambda {a_{2}}^{1-\mathscr{P}}}{(\mu +1)}+ \frac{\vartheta \mathscr{Q}}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr] . \end{aligned} $$
(3.22)

Proof

Use the weighted \(\mathscr{AM}-\mathscr{GM}\) inequality

$$ \begin{aligned} c^{\lambda }d^{\vartheta }\leq {\lambda }c+{ \vartheta }d,\quad c,d \geq 0, \lambda , \vartheta >0, \lambda +\vartheta =1. \end{aligned} $$
(3.23)

Using Lemma 3.1 and by using the n-polynomial \(\mathscr{P}\)-convexity of \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) we find

$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}}) +({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1} \bigl[\zeta ^{\mu } \bigl( \zeta {a_{1}}^{ \mathscr{P}} +(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr]^{ \lambda } \bigl[ \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \bigr]^{ \vartheta } \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} ]^{ \lambda } \bigl[ \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \bigr]^{\vartheta } \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \biggl[ \int _{0}^{1}\lambda \zeta ^{\mu } \bigl\vert \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert \,d\zeta \\ &\quad\quad{} + \int _{0}^{1} \vartheta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \biggr] \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \biggl[ \int _{0}^{1}\lambda \zeta ^{\mu } \bigl\vert \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert \,d\zeta \\ &\quad\quad{} + \int _{0}^{1} \vartheta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \biggr] \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \Biggl[ \int _{0}^{1}{a_{1}}^{1-\mathscr{P}} \lambda \zeta ^{\mu } \,d\zeta + \int _{0}^{1}\vartheta \Biggl\vert \frac{1}{n} \sum_{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert \\ &\quad\quad{} + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-\zeta ^{ \theta } \bigr] \big\vert \psi ^{( \mu +1)}(z) \big\vert \Biggr\vert \,d\zeta \Biggr] \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \Biggl[ \int _{0}^{1}{a_{1}}^{1-\mathscr{P}} \lambda \zeta ^{\mu } \,d\zeta + \int _{0}^{1}\vartheta \Biggl\vert \frac{1}{n} \sum_{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{2}}) \bigr\vert \\ &\quad\quad{} + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-\zeta ^{ \theta } \bigr] \big\vert \psi ^{( \mu +1)}(z) \big\vert \Biggr\vert \,d\zeta \Biggr] \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[\frac{\lambda {a_{1}}^{1-\mathscr{P}}}{(\mu +1)}+ \frac{\vartheta \mathscr{Q}}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr]. \end{aligned}$$
(3.24)

Continuing in the same fashion, we can prove the second part. □

Theorem 3.6

For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) a n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):

$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \bigl(\Lambda _{1}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{\theta =1}^{n} \frac{\theta }{\theta +1} \Biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{2}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{\theta ^{2}+2\theta -1}{(\theta +2)(\theta +1)} \Biggr)^{1/ \vartheta } \Biggr] \end{aligned}$$
(3.25)

and the following inequality holds for all \(z\in ({a_{1}}, {a_{2}})\) and \(\mathscr{P}\in (-\infty , 0) \cup (0,1)\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}- \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \bigl(\Lambda _{3}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{\theta =1}^{n} \frac{\theta }{\theta +1} \Biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{4}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{\theta ^{2}+2\theta -1}{(\theta +2)(\theta +1)} \Biggr)^{1/ \vartheta } \Biggr]. \end{aligned} , $$
(3.26)

Here

$$ \begin{aligned} & \Lambda _{1}({a_{1}},z; \mathscr{P})= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ]}{{a_{1}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \\ & \Lambda _{2}({a_{1}},z;\mathscr{P})= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2, \lambda \mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (- \infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2, \lambda \mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ]}{{a_{1}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (1, \infty ), \end{cases}\displaystyle \\ & \Lambda _{3}({a_{2}},z;\mathscr{P})= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-({a_{2}}/z)^{\mathscr{P}} ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-(z/{a_{2}})^{\mathscr{P}} ]}{{a_{2}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \end{aligned} $$
(3.27)

and

$$ \Lambda _{4}({a_{2}},z;\mathscr{P}) = \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-({a_{2}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (- \infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-(z/{a_{2}})^{\mathscr{P}} ) ]}{{a_{2}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (1, \infty ). \end{cases} $$
(3.28)

Proof

We use Lemma 3.1 to prove the first part of the inequality and we use the Hölder–İşcan inequality to find

$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \biggl( \int _{0}^{1}\zeta ^{\lambda \mu }(1-\zeta ) \bigl( \zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \biggl( \int _{0}^{1}\zeta ^{\lambda \mu +1} \bigl( \zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta ) z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{ \zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \biggl( \int _{0}^{1} \zeta ^{\lambda \mu }(1-\zeta ) \bigl( \zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \lambda (\frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1/ \lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \biggl( \int _{0}^{1}\zeta ^{\lambda \mu +1} \bigl( \zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{ \lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})}\,d\zeta \biggr)^{1/ \lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{ \zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}\times \biggl[ \bigl(\Lambda _{1}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \biggl( \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{2}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \biggl( \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}\times \biggl[ \bigl(\Lambda _{3}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \biggl( \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{4}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \biggl( \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr]. \end{aligned}$$
(3.29)

As \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) is n-polynomial \(\mathscr{P} \)-convex and \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z \in [{a_{1}}, {a_{2}}]\), we find

$$\begin{aligned} & \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}(1-\zeta ) \Biggl[ \frac{1}{n} \sum_{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta }+\frac{1}{n}\sum_{ \theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{\vartheta } \Biggr]\,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n} \sum_{\theta =1}^{n} \int _{0}^{1} \bigl[2(1-\zeta )-(1-\zeta )^{\theta +1}-\zeta ^{ \theta }(1-\zeta ) \bigr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{\theta }{\theta +1}, \\ & \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}\zeta \Biggl[\frac{1}{n} \sum _{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta }+\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl[2\zeta -\zeta (1-\zeta )^{\theta }-\zeta ^{ \theta +1} \bigr]\,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{\theta ^{2}+2\theta -1}{(\theta +2)(\theta +1)}. \end{aligned}$$
(3.30)

Similarly, we obtain

$$ \begin{aligned} & \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \leq \frac{\mathscr{Q}^{\vartheta }}{n} \sum_{\theta =1}^{n} \frac{\theta }{\theta +1}, \\ & \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \leq \frac{\mathscr{Q}^{\vartheta }}{n} \sum_{\theta =1}^{n} \frac{\theta ^{2}+2\theta -1}{(\theta +2)(\theta +1)}. \end{aligned} $$
(3.31)

We have the result

$$\begin{aligned}& \begin{aligned} \Lambda _{1}({a_{1}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \lambda \mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \\ &= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-({a_{1}}/z)^{\mathscr{P}} ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-(z/{a_{1}})^{\mathscr{P}} ]}{{a_{1}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.32)
$$\begin{aligned}& \begin{aligned} \Lambda _{2}({a_{1}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \lambda \mu +1} \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{\lambda (\frac{1-\mathscr{P}}{\mathscr{P}})} \,d \zeta \\ &= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (- \infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ]}{{a_{1}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (1, \infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.33)
$$\begin{aligned}& \begin{aligned} \Lambda _{3}({a_{2}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \lambda \mu }(1-\zeta ) \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \\ &= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-({a_{2}}/z)^{\mathscr{P}} ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-(z/{a_{2}})^{\mathscr{P}} ]}{{a_{2}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.34)
$$\begin{aligned}& \begin{aligned} \Lambda _{4}({a_{2}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \lambda \mu +1} \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{\lambda (\frac{1-\mathscr{P}}{\mathscr{P}})} \,d \zeta \\ &= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-({a_{2}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (- \infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-(z/{a_{2}})^{\mathscr{P}} ) ]}{{a_{2}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (1, \infty ). \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.35)

 □

Theorem 3.7

For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ } \) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) an n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):

$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}- \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \bigl(\Lambda _{1}^{*}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{\theta =1}^{n} \mathrm{T} _{1}({a_{1}},z; \mathscr{P}) \Biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{2}^{*}({a_{1}},z; \mathscr{P}) \bigr)^{1/\lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{ \theta =1}^{n}\mathrm{T} _{2}({a_{1}},z; \mathscr{P}) \Biggr)^{1/ \vartheta } \Biggr] \end{aligned}$$
(3.36)

and the following inequality holds for all \(z\in ({a_{1}}, {a_{2}})\) and \(\mathscr{P}\in (-\infty , 0)\cup (0,1)\):

$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}- \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \bigl(\Lambda _{3}^{*}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{\theta =1}^{n} \mathrm{T} _{3}({a_{2}},z; \mathscr{P}) \Biggr)^{1/\vartheta } \\ & \qquad {} + \bigl(\Lambda _{4}^{*}({a_{2}},z; \mathscr{P}) \bigr)^{1/\lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{ \theta =1}^{n}\mathrm{T} _{4}({a_{2}},z; \mathscr{P}) \Biggr)^{1/ \vartheta } \Biggr] .\end{aligned} $$
(3.37)

Here

$$\begin{aligned} & \mathrm{T} _{1}({a_{1}}z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{(\mathscr{P}-1)}} [ \frac{2}{(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +1,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\theta +\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ] \\ \quad {}-\mathbb{B}(\mu +\theta +1,2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +\theta +1,\theta +\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{(\mathscr{P}-1)}} [ \frac{2}{(\mu +1)(\mu +2)} \mathscr{F}_{1} (1-1/\mathscr{P},\mu +1, \mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +1,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\theta +\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ] \\ \quad {}-\mathbb{B}(\mu +\theta +1,2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +\theta +1,\theta +\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \\ & \mathrm{T} _{2}({a_{1}}z;\mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{(\mathscr{P}-1)}} [\frac{2}{(\mu +2)} \mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +2,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\theta +\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ] \\ \quad {}-\frac{1}{\mu +\theta +2}{}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P}, \mu +\theta +2,\theta +\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) ], \\ \quad \mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{(\mathscr{P}-1)}} [\frac{2}{(\mu +2)} \mathscr{F}_{1} (1-1/\mathscr{P},\mu +2,\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +2,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\theta +\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ] \\ \quad {}-\frac{1}{\mu +\theta +2}{}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P}, \mu +\theta +2,\theta +\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) ], \\ \quad \mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \\ & \Lambda _{1}^{*}({a_{1}},z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +1)(\mu +2)} \mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +1,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{\mathscr{P}-1}(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +1,\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \\ & \Lambda _{2}^{*}({a_{1}},z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +2,a\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \\ & \Lambda _{3}^{*}({a_{2}},z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +1)(\mu +2)} \mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +1,\mu +3,1-({a_{2}}/z)^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{2}}^{\mathscr{P}-1}(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +1,\mu +3,1-(z/{a_{2}})^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \end{aligned}$$
(3.38)

and

$$ \Lambda _{4}^{*}({a_{2}},z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-({a_{2}}/z)^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{2}}^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-(z/{a_{2}})^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ). \end{cases}\displaystyle . $$
(3.39)

Proof

By using Lemma 3.1 and the improved power-mean inequality

$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu } \psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}\times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl( \zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad\quad{}\times \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl(\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl( \zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta ) z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad\quad{}\times \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta ) z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{2}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \bigl(\Lambda _{1}^{*}({a_{1}},z; \mathscr{P}) \bigr)^{1-1/ \lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu }(1- \zeta ) \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad\quad{}\times \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{2}^{*}({a_{1}},z; \mathscr{P}) \bigr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \bigl(\Lambda _{3}^{*}({a_{2}},z; \mathscr{P}) \bigr)^{1-1/ \lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl( \zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad\quad{}\times \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{4}^{*}({a_{2}},z; \mathscr{P}) \bigr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr]. \end{aligned}$$
(3.40)

As \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) is n-polynomial \(\mathscr{P} \)-convex and \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}}, {a_{2}}]\), we have

$$\begin{aligned} \begin{aligned} & \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ & \qquad {} \times \Biggl[\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-(1- \zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta } +\frac{1}{n}\sum _{\theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{(\frac{1-\mathscr{P}}{\mathscr{P}})} \bigl[2 \zeta ^{\mu }(1- \zeta )-\zeta ^{\mu }(1-\zeta )^{ \theta +1}-\zeta ^{ \mu +\theta }(1- \zeta ) \bigr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \mathrm{T} _{1}({a_{1}}z;\mathscr{P}), \end{aligned} \end{aligned}$$
(3.41)

where

$$\begin{aligned} \mathrm{T} _{1}({a_{1}}z; \mathscr{P})&:= \int _{0}^{1} \bigl( \zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad {}\times \bigl[2\zeta ^{\mu }(1-\zeta )-\zeta ^{ \mu }(1- \zeta )^{\theta +1}-\zeta ^{\mu +\theta }(1-\zeta ) \bigr]\,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{(\mathscr{P}-1)}} [ \frac{2}{(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} (1-1/\mathscr{P}, \mu +1,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +1,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\theta +\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ] \\ \quad {}-\mathbb{B}(\mu +\theta +1,2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +\theta +1,\theta +\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{(\mathscr{P}-1)}} [ \frac{2}{(\mu +1)(\mu +2)} \mathscr{F}_{1} (1-1/\mathscr{P},\mu +1, \mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +1,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\theta +\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ] \\ \quad {}-\mathbb{B}(\mu +\theta +1,2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +\theta +1,\theta +\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ). \end{cases}\displaystyle \end{aligned}$$
(3.42)

Similarly, we obtain

$$ \begin{aligned} & \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ & \qquad {} \times \Biggl[\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-(1- \zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta }+ \frac{1}{n}\sum _{\theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{(\frac{1-\mathscr{P}}{\mathscr{P}})} \bigl[2 \zeta ^{\mu +1} - \zeta ^{\mu +1}(1-\zeta )^{\theta }-\zeta ^{\mu + \theta +1} \bigr]\,d \zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \mathrm{T} _{2}({a_{1}}z;\mathscr{P}), \end{aligned} $$
(3.43)

where

$$\begin{aligned} \begin{aligned} \mathrm{T} _{2}({a_{1}}z; \mathscr{P}) :={}& \int _{0}^{1} \bigl( \zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl[2\zeta ^{\mu +1}- \zeta ^{\mu +1}(1- \zeta )^{\theta }-\zeta ^{\mu +\theta +1} \bigr]\,d \zeta \\ ={}& \textstyle\begin{cases} \frac{1}{z^{(\mathscr{P}-1)}} [\frac{2}{(\mu +2)} \mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +2,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\theta +\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ] \\ \quad {}- \frac{1}{\mu +\theta +2}{}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P}, \mu +\theta +2,\theta +\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) ], \\ \quad \mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{(\mathscr{P}-1)}} [\frac{2}{(\mu +2)} \mathscr{F}_{1} (1-1/\mathscr{P},\mu +2,\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +2,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\theta +\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ] \\ \quad {}-\frac{1}{\mu +\theta +2}{}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P}, \mu +\theta +2,\theta +\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) ], \\ \quad \mathscr{P} \in (1,\infty ). \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.44)

We obtain \(\mathrm{T} _{3}({a_{2}},z;\mathscr{P})\) and \(\mathrm{T} _{4}({a_{2}},z;\mathscr{P})\) by replacing \({a_{1}}\) into \({a_{2}}\) in the above results and using the facts

$$\begin{aligned}& \begin{aligned} \Lambda _{1}^{*}({a_{1}},z; \mathscr{P})&:= \int _{0}^{1} \zeta ^{ \mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{(\frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +1)(\mu +2)} \mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +1,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{\mathscr{P}-1}(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +1,\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.45)
$$\begin{aligned}& \begin{aligned} \Lambda _{2}^{*}({a_{1}},z; \mathscr{P})&:= \int _{0}^{1} \zeta ^{ \mu +1} \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})}\,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +2,a\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.46)
$$\begin{aligned}& \begin{aligned} \Lambda _{3}^{*}({a_{2}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \mu }(1-\zeta ) \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{(\frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +1)(\mu +2)} \mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +1,\mu +3,1-({a_{2}}/z)^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{2}}^{\mathscr{P}-1}(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +1,\mu +3,1-(z/{a_{2}})^{ \mathscr{P}} ), \\ \quad \mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.47)

and

$$ \begin{aligned} \Lambda _{4}^{*}({a_{2}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \mu +1} \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-({a_{2}}/z)^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{2}}^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-(z/{a_{2}})^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \end{aligned} $$
(3.48)

which completes the proof. □

Special bi-variate means

Let \(\lambda _{1}, \lambda _{2}, w_{1}, w_{2}>0\). Then the arithmetic mean \(\mathcal{A}(\lambda _{1}, \lambda _{2})\), harmonic mean \(\mathcal{H}(\lambda _{1}, \lambda _{2})\) and weighted arithmetic mean \(\mathcal{B}(\lambda _{1}, \lambda _{2};w_{1};w_{2})\) are defined by \(\mathcal{A}(\lambda _{1}, \lambda _{2})= \frac{\lambda _{1}+\lambda _{2}}{2}\), \(\mathcal{H}( \lambda _{1}, \lambda _{2})= \frac{2\lambda _{1}\lambda _{2}}{\lambda _{1}+\lambda _{2}}\) and \({\mathcal{B}}({\lambda _{1}}, {\lambda _{2}};{w_{1}};{w_{2}}) = \frac{{{\lambda _{1}}{w_{1}} + {\lambda _{2}}{w_{2}}}}{{{w_{1}} + {w_{2}}}}\). The given propositions can be obtained by making some proper substitutions in Theorem 3.1.

Proposition 4.1

The inequality

$$\begin{aligned} \begin{gathered} \biggl\vert \frac{2}{3(\lambda _{2}-\lambda _{1})} \bigl[ \bigl(2\mathcal{B} (\sqrt{\lambda _{1}}, \sqrt{\lambda _{2}})-\lambda _{1} \bigr)^{3}- \bigl(2 \mathcal{B}(\sqrt{\lambda _{1}}, \sqrt{ \lambda _{2}})-\lambda _{2} \bigr)^{3} \bigr] \\ \quad{} - \bigl[2\mathcal{B}(\sqrt{\lambda _{1}}, \sqrt{\lambda _{2}})-\mathcal{B}( \lambda _{1}, \lambda _{2}) \bigr]^{2} \biggr\vert \end{gathered} \end{aligned}$$

holds for all \(\lambda _{2}>\lambda _{1}>0 \).

Proposition 4.2

The inequality

$$\begin{aligned} & \biggl\vert \frac{2}{{(m + 1)({\lambda _{2}} - {\lambda _{1}})}} \bigl[ { \bigl(2{ \mathscr{B}} (\sqrt{{ \lambda _{1}}} , \sqrt{{\lambda _{2}}} ) - { \lambda _{1}} \bigr)^{m + 1}} \\ & \quad\quad{} - { \bigl(2{\mathscr{B}}(\sqrt{{\lambda _{1}}} , \sqrt{{\lambda _{2}}} ) - {\lambda _{2}} \bigr)^{m + 1}} \bigr] - { \bigl[2{\mathscr{B}}(\sqrt{{\lambda _{1}}} , \sqrt{{\lambda _{2}}} ) - {\mathscr{B}}({\lambda _{1}},{\lambda _{2}}) \bigr]^{m}} \biggr\vert \\ & \quad \le \frac{{3m({\lambda _{2}} - {\lambda _{1}})}}{4}{\mathscr{B}} ( { \bigl({\mathscr{B}} \bigl( \lambda _{1}^{\frac{{m - 1}}{2}}, {d^{ \frac{{m - 2}}{{{2^{2}}}}}} \bigr), {\mathscr{B}} \bigl( \lambda _{2}^{m - 2}, \lambda _{2}^{m - 1} \bigr)} \bigr) \end{aligned}$$

holds for all \(\lambda _{2}>\lambda _{1}>0\) and \(m\in \mathbb{N}\) with \(m\geq 2\).

Proposition 4.3

The inequality

$$\begin{aligned} & \biggl\vert \frac{2}{{(m + 1)({\lambda _{2}} - {\lambda _{1}})}} \bigl[ { \bigl(2{ \mathscr{B}} (\sqrt{{ \lambda _{1}}} , \sqrt{{\lambda _{2}}} ) - { \lambda _{1}} \bigr)^{m + 1}} \\ &\quad\quad{} - { \bigl(2{\mathscr{B}}(\sqrt{{\lambda _{1}}} , \sqrt{{\lambda _{2}}} ) - {\lambda _{2}} \bigr)^{m + 1}} \bigr] - { \bigl[2{\mathscr{B}}(\sqrt{{\lambda _{1}}} , \sqrt{{\lambda _{2}}} ) - {\mathscr{B}}({\lambda _{1}},{\lambda _{2}}) \bigr]^{m}} \biggr\vert \\ & \quad \le \frac{{3m({\lambda _{2}} - {\lambda _{1}})}}{4}{\mathscr{B}} ( { \bigl({\mathscr{B}} \bigl( \lambda _{1}^{\frac{{m - 1}}{2}}, \lambda _{2}^{ \frac{{m - 1}}{2}} \bigr), { \mathscr{B}} \bigl(\lambda _{2}^{m - 2},\lambda _{2}^{m - 1} \bigr)} \bigr) \end{aligned}$$

holds for all \(\lambda _{2}>\lambda _{1}>0\) and \(m\in \mathbb{N}\) with \(m\geq 2\).

Concluding remarks

In this article, we establish novel Ostrowski-type inequalities for n-polynomial \(\mathscr{P} \)-convex functions. To the best of our knowledge, these results are new in the literature. Since convex functions have immense applications in many mathematical areas, we hope that our new developments can be applied to special functions, and in convex analysis, quantum analysis, post-quantum analysis, related optimization theory, mathematical inequalities and that they may stimulate further research in various areas of pure and applied sciences. In the end, we have given some applications.

Availability of data and materials

Not applicable.

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Acknowledgements

The authors would like to express their sincere thanks to the support of National Natural Science Foundation of China.

Funding

The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 61673169).

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yu-Ming Chu.

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Naz, S., Naeem, M.N. & Chu, YM. Ostrowski-type inequalities for n-polynomial \(\mathscr{P}\)-convex function for k-fractional Hilfer–Katugampola derivative. J Inequal Appl 2021, 117 (2021). https://doi.org/10.1186/s13660-021-02657-0

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MSC

  • 26D10
  • 26D15
  • 90C23

Keywords

  • Generalized k-fractional Hilfer–Katugampola derivative
  • Generalized k-Riemann–Liouville fractional integral
  • Convex function
  • Hermite–Hadamard inequality
  • Ostrowski inequality