Perturbed trapezoid inequalities for <em>n</em> th order differentiable convex functions and their applications

Duygu Dönmez Demir; Gülsüm Şanal; Duygu Dönmez Demir; Gülsüm Şanal

doi:10.3934/math.2020352

AIMS Mathematics

2020, Volume 5, Issue 6: 5495-5509. doi: 10.3934/math.2020352

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Research article

Perturbed trapezoid inequalities for n th order differentiable convex functions and their applications

Duygu Dönmez Demir ^,,
Gülsüm Şanal

Department of Mathematics, Manisa Celal Bayar University, Manisa, Turkey

Received: 05 February 2020 Accepted: 18 May 2020 Published: 28 June 2020
MSC : 39B62, 52A41

In this study, we introduce a new general identity for n th order differentiable functions. Also, we establish some new inequalities regarding general perturbed trapezoid inequality for the functions whose the absolute values of n th derivatives are convex. Finally, some applications for special means are provided.

Keywords:

convex function,
perturbed trapezoid inequalities

Citation: Duygu Dönmez Demir, Gülsüm Şanal. Perturbed trapezoid inequalities for n th order differentiable convex functions and their applications[J]. AIMS Mathematics, 2020, 5(6): 5495-5509. doi: 10.3934/math.2020352

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Abstract

1. Introduction

Many researchers present a large number of studies to improve and generalize classical Hermite-Hadamard inequality. This double inequality suggests that the mean value of a continuous convex function $g: \left[ a, b\right] \subseteq \mathbb{R} \rightarrow \mathbb{R}$ lies between the value of $g$ in the midpoint of the interval $\left[ a, b \right]$ and the arithmetic mean of the values of $g$ at the endpoints of this interval such that

$\begin{equation} g\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\int_{a}^{b}g\left( x\right) dx\leq \frac{g\left( a\right) +g\left( b\right) }{2}. \end{equation}$

(1.1)

In addition, each side of the mentioned inequality characterizes convexity in the sense that a real-valued continuous function $g$ defined on an interval $I$ is convex if its restriction to each compact subinterval $\left[ a, b\right] \subset I$ hold both inequalities. If $g$ is a concave function, then the inequality is interchanged 1.2 ^[3,23]. In the literature, Hermite-Hadamard inequality is frequently preferred because of its importance in nonlinear analysis. Recently, Jain et al. ^[10] established some new inequalities related to Hermite-Hadamard inequality for the functions whose absolute values of second derivatives are $log-$ convex. Mehrez and Agarwal ^[13] introduced new Hermit-Hadamard type integral inequalities for convex functions. Mo-hammed ^[16] presented some new Hermite-Hadamard inequalities for $MT-$ convex functions. Besides, some new integral inequalities for the logarithmically $p-$ preinvex functions via generalized beta function were established by Mohammed ^[17].

Many authors introduced a large number of studies to generalize various integral inequalities. Cerone et al. ^[4] presented the generalized trapezoid inequality. Ujević ^[27] derived some new perturbations of the trapezoid inequality. Drogomir et al. ^[6] improved quasi-trapezoid quadrature formula by using some well-known classical inequalities. Cerone ^[5] obtained explicit bounds for perturbed trapezoidal rules. Liu and Park ^[11] suggested some perturbed versions of generalized trapezoid inequality. On the other hand, some integral inequalities for the convex functions have been frequently investigated by many researchers. Sarikaya and Aktan ^[24] intoduced the generalization of some integral inequalities for convex functions. Tunç and Şanal ^[26] established some perturbed trapezoid inequalities for twice differentiable convex, $s-$ convex and $tgs-$ convex functions. Ardıc ^[1] presented some integral inequalities such as Hölder, Hermite-Hadamard and Jensen integral inequality for $n$ times differentiable convex functions.

The studies in recent years have focused on the fractional integral inequalities for convex functions. Agarwal et al. ^[2] established some fractional integral inequalities via new Pólya-Szegö type integral inequalities. Fernandez and Mohammed ^[9] used the fractional integrals to obtain the Hermite-Hadamard inequality and related results. Mohammed ^[15] introduced some new integral inequalities by using the $\left(k, h\right)$ , $\left(k, s\right)$ -Riemann Liouville fractional integrals. The author presented new Hermite-Hadamard's type inequalities for Riemann Liouville fractional integrals of convex function ^[18]. Besides, Hermite-Hadamard's type inequalities have been obtained via the fractional integrals for different type convex functions ^{[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]}.

The main aim of the present study is to obtain some new inequalities related to general perturbed trapezoid inequality. The considered classes of functions consist of the functions whose n th derivatives of absolute values are convex.

Definition 1. ^[14] A function $g:I\subset \mathbb{R} \rightarrow \mathbb{R}$ is said to be convex on $I$ if inequality

$\begin{equation} g\left( ta+\left( 1-t\right) b\right) \leq tg\left( a\right) +\left( 1-t\right) g\left( b\right) \end{equation}$

(1.2)

holds for all $a, b\in I$ and $t\in \left[ 0, 1\right]$ . We say that $g$ is concave if $(-g)$ is convex. For numerical integration, the trapezoid inequality is introduced as

$\begin{equation} \left\vert \int_{a}^{b}g\left( x\right) dx-\frac{1}{2}\left( b-a\right) \left( g\left( a\right) +g\left( b\right) \right) \right\vert \leq \frac{1}{ 12}M_{2}\left( b-a\right) ^{3} \end{equation}$

(1.3)

where $g:\left[ a, b\right] \rightarrow \mathbb{R}$ is supposed to be twice differentiable on the interval $\left(a, b\right)$ , with the second derivative bounded on $\left(a, b\right)$ by $M_{2} = \sup_{x\in \left(a, b\right) }\left\vert g^{\prime \prime }\left(x\right) \right\vert < +\infty$

(^{[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]}).

Theorem 1. Grüss inequality : Let $g$ and $z$ to be two functions defined and integrable on $\left[ a, b\right]$ . If $k\leq g\left(x\right) \leq l$ and $m\leq z\left(x\right) \leq n$ is to be $\forall x\in \left[ a, b \right]$ and for constants $k, l, m, n\in \mathbb{R}$ , then

$\begin{equation} \left\vert \frac{1}{b-a}\int_{a}^{b}g\left( x\right) z\left( x\right) dx- \frac{1}{b-a}\int_{a}^{b}g\left( x\right) dx\frac{1}{b-a}\int_{a}^{b}z\left( x\right) dx\right\vert \leq \frac{1}{4}\left( l-k\right) \left( n-m\right) . \end{equation}$

(1.4)

The above inequality is held where $\frac{1}{4}$ is the best constant ^[7]. For the perturbed trapezoid inequality, the inequality obtained by the application of the Grüss inequality is given as

$\begin{eqnarray} &&\left\vert \int_{a}^{b}g\left( x\right) dx-\frac{1}{2}\left( b-a\right) \left( g\left( a\right) +g\left( b\right) \right) +\frac{1}{12}\left( b-a\right) ^{2}\left( g^{\prime }\left( b\right) -g^{\prime }\left( a\right) \right) \right\vert \\ &\leq &\frac{1}{32}\left( \Gamma _{2}-\gamma _{2}\right) \left( b-a\right) ^{3} \end{eqnarray}$

(1.5)

by Dragomir et al. ^[6] where $g$ is supposed to be twice differentiable on the interval $\left(a, b\right)$ with the second derivative bounded on $\left(a, b\right)$ by $\Gamma _{2} = \sup_{x\in \left(a, b\right) }g^{\prime \prime }\left(x\right) < +\infty$ and $\gamma _{2} = \inf_{x\in \left(a, b\right) }g^{\prime \prime }\left(x\right) > -\infty$ .

In the light of this information, we establish some inequalities for n th order differentiable convex functions.

Lemma 1. ^[26] Let $g$ $:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ , $a, b\in I^{\circ }$ with $a < b$ . If $g^{\prime \prime }\in L[a, b]$ , then one obtains

$\begin{eqnarray} &&\int_{a}^{b}g\left( x\right) dx-\frac{1}{2}\left( b-a\right) \left( g\left( a\right) +g\left( b\right) \right) +\frac{5}{4}\left( b-a\right) ^{2}\left( g^{\prime }\left( b\right) -g^{\prime }\left( a\right) \right) \\ & = &\frac{\left( b-a\right) ^{3}}{4}\int_{0}^{1}\left( t+1\right) ^{2}\left[ g^{\prime \prime }\left( ta+\left( 1-t\right) b\right) +g^{\prime \prime }\left( tb+\left( 1-t\right) a\right) \right] dt. \end{eqnarray}$

(1.6)

Theorem 2. ^[12] Minkowski Inequality: Let $g^{p}$ , $z^{p}$ and $\left(g+z\right) ^{p}$ be integrable functions on $\left[ a, b \right]$ . If $p > 1$ , then

$\begin{equation} \left[ \int_{a}^{b}\left\vert g\left( x\right) +z\left( x\right) \right\vert ^{p}dx\right] ^{\frac{1}{p}}\leq \left[ \int_{a}^{b}\left\vert g\left( x\right) \right\vert ^{p}dx\right] ^{\frac{1}{p}}+\left[ \int_{a}^{b}\left \vert z\left( x\right) \right\vert ^{p}dx\right] ^{\frac{1}{p}}. \end{equation}$

(1.7)

Similarly, if $p > 1$ and $a_{k}$ , $b_{k} > 0$ , then Minkowski sum inequality is expressed as

$\begin{equation} \left[ \sum\limits_{k = 1}^{n}\left\vert a_{k}+b_{k}\right\vert ^{p}\right] ^{\frac{1 }{p}}\leq \left[ \sum\limits_{k = 1}^{n}\left\vert a_{k}\right\vert ^{p}\right] ^{ \frac{1}{p}}+\left[ \sum\limits_{k = 1}^{n}\left\vert b_{k}\right\vert ^{p}\right] ^{ \frac{1}{p}}. \end{equation}$

(1.8)

If the sequences $a_{1}$ , $a_{2}$ , ... and $b_{1}$ , $b_{2}$ , ... are proportional, the inequality is provided. We will use the following notations and conventions throughout this article. Let us consider as $I = \left[ 0, \infty \right) \subset \mathbb{R} = \left(-\infty, +\infty \right)$ and $a, b\in I$ with $0 < a < b$ and $g^{\left(n\right) }\in L\left[ a, b\right]$ and

$\begin{eqnarray*} A\left( a,b\right) = \frac{a+b}{2},G\left( a,b\right) = \sqrt{ab}, H\left( a,b\right) = \frac{2ab}{a+b}, \\ L\left( a,b\right) = \frac{b-a}{\ln b-\ln a},a\neq b, L_{p}\left( a,b\right) , \\ L_{p} = L_{p}\left( a,b\right) = \left\{ \begin{array}{ccc} a , a = b \\ \left[ \frac{b^{p+1}-a^{p+1}}{\left( p+1\right) \left( b-a\right) }\right] ^{ \frac{1}{p}} , a\neq b \end{array} \right. \;\; a,b\geq 0 \end{eqnarray*}$

are the arithmetic mean, geometric mean, harmonic mean, logarithmic mean, generalized $p$ -logarithmic mean for $a, b > 0,$ respectively ^[8].

In this study, we introduce some results related to the perturbed trapezoid inequality and prove some applications for special means of real numbers.

2. Main results

Lemma 2. Let $g:I^{\circ }\subseteq \mathbb{R} \rightarrow \mathbb{R}$ be $n$ times differentiable mapping on $I^{\circ }$ , $a, b\in I^{\circ }$ with $a < b$ $\$ where $n$ is even number. If $g^{\left(n\right) }\in L[a, b]$ , then the following equality is obtained:

$\begin{eqnarray} &&\frac{1}{b-a}\int\limits_{b}^{a}g\left( x\right) dx-\frac{g\left( a\right) +g\left( b\right) }{2}+\cdots \\ &&+\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) \left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] }{2.n!.a_{n}} \\ &&\times \left( g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right) \\ &&+\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+4.a_{2}\right] }{2.n!.a_{n}} \\ &&\times \left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \\ &&-\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] }{ 2.n!.a_{n}}\times \left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] \\ &&+\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+\cdots +a_{1}+2a_{0}\right] }{ 2.n!.a_{n}}\times \left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] \\ & = &\frac{\left( b-a\right) ^{n}}{2.n!.a_{n}} \\ &&\times \int\limits_{0}^{1}\left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) \left[ g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) +g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right] dt \end{eqnarray}$

(2.1)

Proof. If the right-hand side of the equality is considered and the integration by parts is applied, then one obtains

$\begin{eqnarray*} I_{1} & = &\int\limits_{0}^{1}\left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) dt \\ & = &\left. \left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) \frac{g^{\left( n-1\right) }\left( ta+\left( 1-t\right) b\right) }{a-b}\right\vert _{0}^{1} \\ &&-\frac{1}{\left( a-b\right) }\int\limits_{0}^{1}\left( n.a_{n}t^{n-1}+\cdots +2a_{2}t+a_{1}\right) g^{\left( n-1\right) }\left( ta+\left( 1-t\right) b\right) dt \\ & = &\frac{\left( a_{n}+\cdots +a_{1}+a_{0}\right) }{a-b}g^{\left( n-1\right) }\left( a\right) -\frac{a_{0}}{a-b}g^{\left( n-1\right) }\left( b\right) \\ &&-\frac{1}{\left( a-b\right) }\int\limits_{0}^{1}\left( n.a_{n}t^{n-1}+\cdots +2a_{2}t+a_{1}\right) g^{\left( n-1\right) }\left( ta+\left( 1-t\right) b\right) dt \\ & = &\frac{\left( a_{n}+\cdots +a_{1}+a_{0}\right) }{a-b}g^{\left( n-1\right) }\left( a\right) -\frac{a_{0}}{a-b}g^{\left( n-1\right) }\left( b\right) \\ &&-\frac{n.a_{n}+\cdots +2a_{2}+a_{1}}{\left( a-b\right) ^{2}}g^{\left( n-2\right) }\left( a\right) +\frac{a_{1}}{\left( a-b\right) ^{2}}g^{\left( n-2\right) }\left( b\right) \\ &&+\frac{1}{\left( a-b\right) ^{2}}\times \int\limits_{0}^{1}\left( n.\left( n-1\right) .a_{n}t^{n-2}+\cdots +3.2.a_{3}t+2.a_{2}\right) \times g^{\left( n-2\right) }\left( ta+\left( 1-t\right) b\right) dt \\ & = &\frac{\left( a_{n}+\cdots +a_{1}+a_{0}\right) }{a-b}g^{\left( n-1\right) }\left( a\right) -\frac{a_{0}}{a-b}g^{\left( n-1\right) }\left( b\right) \\ &&-\frac{n.a_{n}+\cdots +2a_{2}+a_{1}}{\left( a-b\right) ^{2}}g^{\left( n-2\right) }\left( a\right) +\frac{a_{1}}{\left( a-b\right) ^{2}}g^{\left( n-2\right) }\left( b\right) \\ &&+\frac{n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+2.a_{2}}{ \left( a-b\right) ^{3}}g^{\left( n-3\right) }\left( a\right) -\frac{2.a_{2}}{ \left( a-b\right) ^{3}}g^{\left( n-3\right) }\left( b\right) \\ &&-\frac{n.\left( n-1\right) .\left( n-2\right) .a_{n}+\cdots +3.2.1.a_{3}}{ \left( a-b\right) ^{4}}g^{\left( n-4\right) }\left( a\right) +\frac{3.2.a_{3} }{\left( a-b\right) ^{4}}g^{\left( n-4\right) }\left( b\right) \\ &&+\cdots -\frac{n!.a_{n}+\left( n-1\right) !.a_{n-1}}{\left( a-b\right) ^{n} }g\left( a\right) +\frac{\left( n-1\right) !.a_{n-1}}{\left( a-b\right) ^{n}} g\left( b\right) \\ &&+\frac{n!a_{n}}{\left( a-b\right) ^{n}}\int\limits_{0}^{1}g\left( ta+\left( 1-t\right) b\right) dt \end{eqnarray*}$

$\begin{eqnarray*} I_{2} & = &\int_{0}^{1}\left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) dt \\ & = &\left. \left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) \frac{g^{\left( n-1\right) }\left( tb+\left( 1-t\right) a\right) }{b-a}\right\vert _{0}^{1} \\ &&-\frac{1}{\left( b-a\right) }\int_{0}^{1}\left( n.a_{n}t^{n-1}+\cdots +2a_{2}t+a_{1}\right) g^{\left( n-1\right) }\left( tb+\left( 1-t\right) a\right) dt \\ & = &\frac{\left( a_{n}+\cdots +a_{1}+a_{0}\right) }{b-a}g^{\left( n-1\right) }\left( b\right) -\frac{a_{0}}{b-a}g^{\left( n-1\right) }\left( a\right) \\ &&-\frac{1}{\left( b-a\right) }\int_{0}^{1}\left( n.a_{n}t^{n-1}+\cdots +2a_{2}t+a_{1}\right) g^{\left( n-1\right) }\left( tb+\left( 1-t\right) a\right) dt \\ & = &\frac{\left( a_{n}+\cdots +a_{1}+a_{0}\right) }{b-a}g^{\left( n-1\right) }\left( b\right) -\frac{a_{0}}{b-a}g^{\left( n-1\right) }\left( a\right) \\ &&-\frac{n.a_{n}+\cdots +2a_{2}+a_{1}}{\left( b-a\right) ^{2}}g^{\left( n-2\right) }\left( b\right) +\frac{a_{1}}{\left( b-a\right) ^{2}}g^{\left( n-2\right) }\left( a\right) \\ &&+\frac{1}{\left( b-a\right) ^{2}}\int_{0}^{1}\left[ n.\left( n-1\right) a_{n}t^{n-2}+\cdots +3.2.a_{3}t+2.a_{2}\right] \times g^{\left( n-2\right) }\left( tb+\left( 1-t\right) a\right) dt \\ & = &\frac{\left( a_{n}+\cdots +a_{1}+a_{0}\right) }{b-a}g^{\left( n-1\right) }\left( b\right) -\frac{a_{0}}{b-a}g^{\left( n-1\right) }\left( a\right) \\ &&-\frac{n.a_{n}+\cdots +2a_{2}+a_{1}}{\left( b-a\right) ^{2}}g^{\left( n-2\right) }\left( b\right) +\frac{a_{1}}{\left( b-a\right) ^{2}}g^{\left( n-2\right) }\left( a\right) \\ &&+\frac{n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+2.a_{2}}{ \left( b-a\right) ^{3}}g^{\left( n-3\right) }\left( b\right) -\frac{2.a_{2}}{ \left( b-a\right) ^{3}}g^{\left( n-3\right) }\left( a\right) \\ &&-\frac{n.\left( n-1\right) .\left( n-2\right) .a_{n}+\cdots +3.2.a_{3}}{ \left( b-a\right) ^{4}}g^{\left( n-4\right) }\left( b\right) +\frac{3.2.a_{3} }{\left( b-a\right) ^{4}}g^{\left( n-4\right) }\left( a\right) \\ &&+\cdots -\frac{n!.a_{n}+\left( n-1\right) !.a_{n-1}}{\left( b-a\right) ^{n} }g\left( b\right) +\frac{\left( n-1\right) !.a_{n-1}}{\left( b-a\right) ^{n}} g\left( a\right) \\ &&+\frac{n!a_{n}}{\left( b-a\right) ^{n}}\int_{0}^{1}g\left( tb+\left( 1-t\right) a\right) dt \end{eqnarray*}$

Summing $\ I_{1}$ and $I_{2}$ , then one obtains

$\begin{eqnarray*} I_{1}+I_{2} & = &\int_{0}^{1}\left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) \left[ g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) +g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right] dt \\ & = &\frac{\left( a_{n}+\cdots +a_{1}+a_{0}\right) }{b-a}\left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] + \frac{a_{0}}{b-a}\left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] \\ &&-\frac{\left( n.a_{n}+\cdots +2a_{2}+a_{1}\right) }{\left( b-a\right) ^{2}} \left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] +\frac{a_{1}}{\left( b-a\right) ^{2}}\left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] \\ &&+\frac{\left( n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2a_{3}+2.a_{2}\right) }{\left( b-a\right) ^{3}}\left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \\ &&+\frac{2.a_{2}}{\left( b-a\right) ^{3}}\left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \\ &&-\frac{\left( n.\left( n-1\right) .\left( n-2\right) a_{n}+\cdots +3.2a_{3}\right) }{\left( b-a\right) ^{4}}\left[ g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right] \\ &&+\frac{3.2.a_{3}}{\left( b-a\right) ^{4}}\left[ g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right] +\cdots \\ &&-\frac{n!.a_{n}+\left( n-1\right) !.a_{n-1}}{\left( b-a\right) ^{n}}\left[ g\left( a\right) +g\left( b\right) \right] +\frac{\left( n-1\right) !.a_{n-1} }{\left( b-a\right) ^{n}}\left[ g\left( a\right) +g\left( b\right) \right] \\ &&+\frac{\left( n\right) !.a_{n}}{\left( b-a\right) ^{n}}\left[ \int_{0}^{1} \left[ g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) +g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right] dt\right] \\ & = &\frac{a_{n}+\cdots +a_{1}+2a_{0}}{b-a}\left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] \\ &&-\frac{n.a_{n}+\cdots +2a_{2}}{\left( b-a\right) ^{2}}\left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] \\ &&+\frac{n.\left( n-1\right) a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+4a_{2}}{ \left( b-a\right) ^{3}}\left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \\ &&-\frac{n.\left( n-1\right) .\left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}}{ \left( b-a\right) ^{4}}\left[ g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right] \\ &&+\cdots -\frac{n!.a_{n}}{\left( b-a\right) ^{n}}\left[ g\left( a\right) +g\left( b\right) \right] +\frac{2.n!.a_{n}}{\left( b-a\right) ^{n+1}} \int_{a}^{b}g\left( x\right) dx \end{eqnarray*}$

$\begin{eqnarray*} &&\frac{1}{b-a}\int_{a}^{b}g\left( x\right) dx-\frac{g\left( a\right) +g\left( b\right) }{2}+\cdots \\ &&+\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) \left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] }{2.n!.a_{n}}\left[ g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right] \\ &&+\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+4.a_{2}\right] }{2.n!.a_{n}}\left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \\ &&-\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] }{ 2.n!.a_{n}}\left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] \\ &&+\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+\cdots +a_{1}+2a_{0}\right] }{ 2.n!.a_{n}}\left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] \\ & = &\frac{\left( b-a\right) ^{n}}{2.n!.a_{n}}\int_{0}^{1}\left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) \times \left[ g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) +g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right] dt \end{eqnarray*}$

Thus, the proof is completed.

Remark 1. Using the change of the variable $x = ta+\left(1-t\right) b$ where $t\in \left[ 0, 1\right]$ , Eq.(2.1) can be written as

$\begin{eqnarray*} &&\frac{1}{b-a}\int_{a}^{b}g\left( x\right) dx-\frac{g\left( a\right) +g\left( b\right) }{2}+\cdots \\ &&+\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) \left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] }{2.n!.a_{n}}\times \left[ g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right] \\ &&+\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+4a_{2}\right] }{2.n!.a_{n}}\times \left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \\ &&-\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] }{ 2.n!.a_{n}}\times \left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] \\ &&+\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+\cdots +a_{1}+2a_{0}\right] }{ 2.n!.a_{n}}\times \left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] \\ & = &\frac{-\left( b-a\right) ^{n-1}}{2.n!.a_{n}}\times \int_{0}^{1}\left( a_{n}\left( \frac{x-b}{a-b}\right) ^{n}+\cdots +a_{1}\left( \frac{x-b}{a-b} \right) +a_{0}\right) \left[ g^{\left( n\right) }\left( x\right) +g^{\left( n\right) }\left( a+b-x\right) \right] dx \end{eqnarray*}$

Theorem 3. Let $g:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ be $n$ times differentiable mapping on $I^{\circ }$ , $a, b\in I^{\circ }$ with $a < b$ where $n$ is even number. If $\left\vert g^{\left(n\right) }\right\vert$ is convex on $[a, b]$ , then the inequality in the following holds:

(2.2)

Proof. From Lemma 2, it is concluded that

$\begin{eqnarray*} &&\left\vert \frac{1}{b-a}\int_{a}^{b}g\left( x\right) dx-\frac{g\left( a\right) +g\left( b\right) }{2}+\cdots \right. \\ &&\left. +\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) \left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] }{2.n!.a_{n}}\times \left[ g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right] \right. \\ &&\left. +\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+2.a_{2}\right] }{2.n!.a_{n}}\times \left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \right. \\ &&\left. -\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] }{2.n!.a_{n}}\times \left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] \right. \\ &&\left. +\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+...+a_{1}+2a_{0}\right] }{2.n!.a_{n}}\times \left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] \right\vert \\ & = &\left\vert \frac{\left( b-a\right) ^{n}}{2.n!.a_{n}}\times \left\{ \int_{0}^{1}\left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) \left[ g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) +g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right] dt\right\} \right\vert \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert } \times \left\{ \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert \left[ \left\vert g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) \right\vert +\left\vert g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right\vert \right] dt\right\} \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert } \\ &&\times \left\{ \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert \left[ t\left\vert g^{\left( n\right) }\left( a\right) \right\vert +\left( 1-t\right) \left\vert g^{\left( n\right) }\left( b\right) \right\vert +t\left\vert g^{\left( n\right) }\left( b\right) \right\vert +\left( 1-t\right) \left\vert g^{\left( n\right) }\left( a\right) \right\vert \right] dt\right\} \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert }\left[ \left\vert g^{\left( n\right) }\left( a\right) \right\vert +\left\vert g^{\left( n\right) }\left( b\right) \right\vert \right] \times \int_{0}^{1} \left[ \left\vert a_{n}t^{n}\right\vert +\cdots +\left\vert a_{1}t\right\vert +\left\vert a_{0}\right\vert \right] dt \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert } \times \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+1}\times \left[ \left\vert g^{\left( n\right) }\left( a\right) \right\vert +\left\vert g^{\left( n\right) }\left( b\right) \right\vert \right] \end{eqnarray*}$

The theorem is proved.

Theorem 4. Let $g:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ be $n$ times differentiable mapping on $I^{\circ }$ , $a, b\in I^{\circ }$ with $a < b$ , and $p > 1$ with $1/p+1/q = 1$ where $n$ is even number. If the mapping $\left\vert g^{\left(n\right) }\right\vert ^{q}$ is convex on $\left[ a, b\right]$ , then we obtain:

(2.3)

Proof. Using Lemma 2, Hölder's integral inequality and Minkowsky's integral inequality, we establish

$\begin{eqnarray} &&\left\vert \frac{1}{b-a}\int_{a}^{b}g\left( x\right) dx-\frac{g\left( a\right) +g\left( b\right) }{2}+\cdots \right. \\ &&\left. +\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) \left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] }{2.n!.a_{n}}\right. \\ &&\left. \times \left[ g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right] \right. \\ &&\left. +\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+2.a_{2}\right] }{2.n!.a_{n}}\right. \\ &&\left. \times \left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \right. \\ &&\left. -\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] }{2.n!.a_{n}}\times \left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] \right. \\ &&\left. +\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+\cdots +a_{1}+2a_{0} \right] }{2.n!.a_{n}}\times \left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] \right\vert \\ & = &\left\vert \frac{\left( b-a\right) ^{n}}{2.n!.a_{n}}\right. \\ &&\left. \times \left\{ \int_{0}^{1}\left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) \left[ g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) +g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right] dt\right\} \right\vert \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert } \\ &&\times \left[ \begin{array}{c} \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert \left\vert g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) \right\vert dt \\ +\int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert \left\vert g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right\vert dt \end{array} \right] \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert } \\ &&\times \left[ \left( \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert ^{p}dt\right) ^{\frac{1}{p}}\times \left( \int_{0}^{1}\left\vert g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) \right\vert ^{q}dt\right) ^{\frac{1}{q}}\right. \\ &&\left. +\left( \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert ^{p}dt\right) ^{\frac{1}{p}}\times \left( \int_{0}^{1}\left\vert g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right\vert ^{q}dt\right) ^{\frac{1}{q}}\right] \\ &\leq &\frac{\left( b-a\right) ^{n}}{n!.\left\vert a_{n}\right\vert }\times \left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{\left( kp+1\right) ^{1/p}}\right] \times \left[ \frac{\left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{q}+\left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{q}}{2}\right] ^{1/q} \end{eqnarray}$

(2.4)

such that $\frac{1}{p}+\frac{1}{q} = 1$ . Considering the convexity of $\left\vert g^{\left(n\right) }\right\vert ^{q}$ , then we find

$\begin{eqnarray} &&\int_{0}^{1}\left\vert g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) \right\vert ^{q}dt \\ &\leq &\int_{0}^{1}\left[ t\left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{q}+\left( 1-t\right) \left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{q}\right] dt = \frac{\left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{q}+\left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{q}}{2} \\ &&\int_{0}^{1}\left\vert g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right\vert ^{q}dt \\ &\leq &\int_{0}^{1}\left[ t\left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{q}+\left( 1-t\right) \left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{q}\right] dt = \frac{\left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{q}+\left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{q}}{2} \end{eqnarray}$

(2.5)

Using the Theorem 2, we have

$\begin{eqnarray} &&\left( \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert ^{p}dt\right) ^{\frac{1}{p}} \\ &\leq &\left[ \int_{0}^{1}\left( a_{n}t^{n}\right) ^{p}dt\right] ^{\frac{1}{p }}+\cdots +\left[ \int_{0}^{1}\left( a_{1}t\right) ^{p}dt\right] ^{\frac{1}{p }}+\left[ \int_{0}^{1}\left( a_{0}\right) ^{p}dt\right] ^{\frac{1}{p}} \\ & = &\frac{\left\vert a_{n}\right\vert }{\left( np+1\right) ^{1/p}}+\frac{ \left\vert a_{n-1}\right\vert }{\left\vert \left( n-1\right) p+1\right\vert ^{1/p}}+\cdots +\frac{\left\vert a_{1}\right\vert }{\left( p+1\right) ^{1/p}} +\left\vert a_{0}\right\vert \\ & = &\left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{\left( kp+1\right) ^{1/p}}\right] \end{eqnarray}$

(2.6)

By using (2.5) and (2.6), the inequality (2.3) is obtained.

Theorem 5. Let $g:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ be $n$ times differentiable mapping on $I^{\circ }$ , $a, b\in I^{\circ }$ with $a < b$ , and $p > 1$ such that $1/p+1/q = 1$ where $n$ is even number. If the mapping $\left\vert g^{\left(n\right) }\right\vert ^{p}$ is convex on $\left[ a, b\right]$ , then the inequality in the following holds:

$\begin{eqnarray} &&\left\vert \frac{1}{b-a}\int_{a}^{b}g\left( x\right) dx-\frac{g\left( a\right) +g\left( b\right) }{2}+\cdots \right. \\ &&\left. +\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) \left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] }{2.n!.a_{n}}\right. \\ &&\left. \times \left[ g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right] \right. \\ &&\left. +\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+4.a_{2}\right] }{2.n!.a_{n}}\right. \\ &&\left. \times \left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \right. \\ &&\left. -\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] }{2.n!.a_{n}}\times \left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] \right. \\ &&\left. +\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+\cdots +a_{1}+2a_{0} \right] }{2.n!.a_{n}}\times \left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] \right\vert \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert }\left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+1}\right] ^{1-\frac{1}{p} } \\ &&\times \left\{ \left( \left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+2}\right] \left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{p}+\left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{\left( k+1\right) \left( k+2\right) }\right] \left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{p}\right) ^{\frac{1}{p} }\right. \\ &&\left. +\left( \left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+2 }\right] \left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{p}+ \left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{\left( k+1\right) \left( k+2\right) }\right] \left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{p}\right) ^{\frac{1}{p}}\right\} \end{eqnarray}$

(2.7)

Proof. Using Lemma 2 and power mean integral inequality, one obtains

$\begin{eqnarray*} &&\left\vert \frac{1}{b-a}\int_{a}^{b}g\left( x\right) dx-\frac{g\left( a\right) +g\left( b\right) }{2}+\cdots \right. \\ &&\left. +\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) \left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] }{2.n!.a_{n}}\right. \\ &&\left. \times \left[ g^{\left( n-4\right) }\left( a\right) +g^{\left( n-4\right) }\left( b\right) \right] \right. \\ &&\left. +\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots +4.3.a_{4}+3.2.a_{3}+4.a_{2}\right] }{2.n!.a_{n}}\right. \\ &&\left. \times \left[ g^{\left( n-3\right) }\left( b\right) -g^{\left( n-3\right) }\left( a\right) \right] \right. \\ &&\left. -\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] }{2.n!.a_{n}}\times \left[ g^{\left( n-2\right) }\left( a\right) +g^{\left( n-2\right) }\left( b\right) \right] \right. \\ &&\left. +\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+\cdots +a_{1}+2a_{0} \right] }{2.n!.a_{n}}\times \left[ g^{\left( n-1\right) }\left( b\right) -g^{\left( n-1\right) }\left( a\right) \right] \right\vert \\ & = &\left\vert \frac{\left( b-a\right) ^{n}}{2.n!.a_{n}}\right. \\ &&\left. \times \left\{ \int_{0}^{1}\left( a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right) \left[ g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) +g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right] dt\right\} \right\vert \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert } \\ &&\times \left[ \begin{array}{c} \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert \left\vert g^{\left( n\right) }\left( ta+\left( 1-t\right) b\right) \right\vert dt \\ +\int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert \left\vert g^{\left( n\right) }\left( tb+\left( 1-t\right) a\right) \right\vert dt \end{array} \right] \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert } \left( \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert dt\right) ^{^{1-\frac{1}{p}}} \\ &&\times \left\{ \left( \begin{array}{c} \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert \\ \left( t\left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{p}+\left( 1-t\right) \left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{p}\right) dt \end{array} \right) ^{\frac{1}{p}}\right. \\ &&\left. +\left( \begin{array}{c} \int_{0}^{1}\left\vert a_{n}t^{n}+\cdots +a_{1}t+a_{0}\right\vert \\ \left( t\left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{p}+\left( 1-t\right) \left\vert g^{n}\left( a\right) \right\vert ^{p}\right) dt \end{array} \right) ^{\frac{1}{p}}\right\} \\ &\leq &\frac{\left( b-a\right) ^{n}}{2.n!.\left\vert a_{n}\right\vert }. \left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+1}\right] ^{1- \frac{1}{p}} \\ &&\times \left\{ \left( \begin{array}{c} \left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+2}\right] .\left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{p} \\ +\left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{\left( k+1\right) \left( k+2\right) }\right] .\left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{p} \end{array} \right) ^{\frac{1}{p}}\right. \\ &&\left. +\left( \begin{array}{c} \left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+2}\right] .\left\vert g^{\left( n\right) }\left( b\right) \right\vert ^{p} \\ +\left[ \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{\left( k+1\right) \left( k+2\right) }\right] .\left\vert g^{\left( n\right) }\left( a\right) \right\vert ^{p} \end{array} \right) ^{\frac{1}{p}}\right\} \end{eqnarray*}$

The proof is completed.

3. Applications to special means

In this section, we consider the results of Section 2 to verify the new proposed inequalities.

Proposition 1. Let $a, b\in \mathbb{R}$ , $0 < a < b$ , $n > 2$ where $n$ is even number. Then, the inequality in the following holds:

$\begin{eqnarray} &&\left\vert L_{n}^{n}\left( a,b\right) -A\left( a^{n},b^{n}\right) +\cdots \right. \\ &&\left. +\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) .\left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] \left( a^{4}+b^{4}\right) }{ 2.4!.a_{n}}\right. \\ &&\left. +\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots +3.2.a_{3}+4.a_{2}\right] \left( b^{3}-a^{3}\right) }{ 2.3!.a_{n}}\right. \\ &&\left. -\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] \left( a^{2}+b^{2}\right) }{2.2!.a_{n}}\right. \\ &&\left. +\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+\cdots +2.a_{0}\right] \left( b-a\right) }{2.a_{n}}\right\vert \\ &\leq &\frac{\left( b-a\right) ^{n}}{\left\vert a_{n}\right\vert }\times \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+1} \end{eqnarray}$

(3.1)

Proof. The proof is clearly obtained from Theorem 3 for $g(x) = x^{n},$ $x\in \mathbb{R}$ .

Proposition 2. Let $a, b\in \mathbb{R}$ , $0 < a < b$ , $n > 2$ where $n$ is even number. For all $p > 1$ , one obtains

$\begin{eqnarray} &&\left\vert L_{n}^{n}\left( a,b\right) -A\left( a^{n},b^{n}\right) +\cdots \right. \\ &&\left. +\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) .\left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] \left( a^{4}+b^{4}\right) }{ 2.4!.a_{n}}\right. \\ &&\left. +\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots +3.2.a_{3}+4.a_{2}\right] \left( b^{3}-a^{3}\right) }{ 2.3!.a_{n}}\right. \\ &&\left. -\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] \left( a^{2}+b^{3}\right) }{2.2!.a_{n}}\right. \\ &&\left. +\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+\cdots +2.a_{0}\right] \left( b-a\right) }{2.a_{n}}\right\vert \\ &\leq &\frac{\left( b-a\right) ^{n}}{\left\vert a_{n}\right\vert }\times \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{\left( kp+1\right) ^{1/p}} \end{eqnarray}$

(3.2)

Proof. The proof is completed from Theorem 4 applied for $g(x) = x^{n},$ $x\in \mathbb{R}$ .

Proposition 3. Let $a, b\in \mathbb{R}$ , $0 < a < b$ , $n > 2$ where $n$ is even number. Then, we obtain for all $p > 1$ ,

$\begin{eqnarray} &&\left\vert L_{n}^{n}\left( a,b\right) -A\left( a^{n},b^{n}\right) +\cdots \right. \\ &&\left. +\frac{-\left( b-a\right) ^{n-4}\left[ n.\left( n-1\right) .\left( n-2\right) .a_{n}+\cdots +4.3.2.a_{4}\right] \left( a^{4}+b^{4}\right) }{ 2.4!.a_{n}}\right. \\ &&\left. +\frac{\left( b-a\right) ^{n-3}\left[ n.\left( n-1\right) .a_{n}+\cdots ++3.2.a_{3}+4.a_{2}\right] \left( b^{3}-a^{3}\right) }{ 2.3!.a_{n}}\right. \\ &&-\frac{\left( b-a\right) ^{n-2}\left[ n.a_{n}+\cdots +2.a_{2}\right] \left( a^{2}+b^{3}\right) }{2.2!.a_{n}} \\ &&\left. +\frac{\left( b-a\right) ^{n-1}\left[ a_{n}+...+2.a_{0}\right] \left( b-a\right) }{2.a_{n}}\right\vert \\ &\leq &\frac{\left( b-a\right) ^{n}}{\left\vert a_{n}\right\vert }\times \left( \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+1}\right) ^{1-1/p}\times \left( \sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{k+2} +\sum\limits_{k = 0}^{n}\frac{\left\vert a_{k}\right\vert }{\left( k+1\right) \left( k+2\right) }\right) ^{\frac{1}{p}} \end{eqnarray}$

(3.3)

Proof. The proof is obtained from Theorem 5 such that $g(x) = x^{n},$ $x\in \left[ a, b\right]$ .

Conflict of interest

The authors declare that there is no conflict of interest.

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AIMS Mathematics

Perturbed trapezoid inequalities for n th order differentiable convex functions and their applications

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2. Main results

3. Applications to special means

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Perturbed trapezoid inequalities for n th order differentiable convex functions and their applications

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1. Introduction

2. Main results

3. Applications to special means

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