Hermite-Hadamard type inequalities for n-times differentiable and preinvex functions

Abstract

In the paper, by creating an integral identity involving an n-times differentiable function, the authors establish some new Hermite-Hadamard type inequalities for preinvex functions and generalize some known results.

MSC:26D15, 26A51, 26B12, 41A55, 49J52.

1 Introduction

Throughout this paper, let $\mathbb{R}=(-\mathrm{\infty },\mathrm{\infty })$ and ℕ denote the set of all positive integers.

Let us recall some definitions of various convex functions.

Definition 1 A function $f:I\subseteq \mathbb{R}\to \mathbb{R}$ is said to be convex if

$$f(\lambda x+(1-\lambda )y)\le \lambda f(x)+(1-\lambda )f(y)$$

(1)

holds for all $x,y\in I$ and $\lambda \in [0,1]$. If the inequality (1) reverses, then f is said to be concave on I.

Definition 2 [1]

A set $S\subseteq {\mathbb{R}}^n$ is said to be invex with respect to the map $\eta :S\times S\to {\mathbb{R}}^n$, if $y+t\eta (x,y)\in S$ for every $x,y\in S$ and $t\in [0,1]$.

It is obvious that every convex set is invex with respect to the map $\eta (x,y)=x-y$, but there exist invex sets which are not convex. See [1], for example.

Definition 3 [1]

Let $S\subseteq {\mathbb{R}}^n$ be an invex set with respect to $\eta :S\times S\to {\mathbb{R}}^n$. For every $x,y\in S$, the η-path $P_{xv}$ joining the points x and $v=x+\eta (y,x)$ is defined by

$$P_{xv}=\{z|z=x+t\eta (y,x),t\in [0,1]\}.$$

(2)

Definition 4 [1]

Let $S\subseteq {\mathbb{R}}^n$ be an invex set with respect to $\eta :S\times S\to {\mathbb{R}}^n$. A function $f:S\to \mathbb{R}$ is said to be preinvex with respect to η, if $f(y+t\eta (x,y))\le tf(x)+(1-t)f(y)$ for every $x,y\in S$ and $t\in [0,1]$.

Every convex function is preinvex with respect to the map $\eta (x,y)=x-y$, but not conversely. For properties and applications of preinvex functions, please refer to [1–3] and closely related references therein.

The most important inequality in the theory of convex functions, the well-known Hermite-Hadamard’s integral inequality, may be stated as follows. If f is a convex function on $[a,b]$, then

$$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x\le \frac{f(a)+f(b)}{2}.$$

(3)

If f is concave on $[a,b]$, then the inequality (3) is reversed.

The inequality (3) has been generalized by many mathematicians. Some of them may be recited as follows.

Theorem 1 [[4], Theorem 2.2]

Let $f:I^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ and $a,b\in I^{\circ }$ with $a<b$. If $|f^{\prime }(x)|$ is convex on $[a,b]$, then

$$|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x|\le \frac{(b-a)[|f^{\prime }(a)|+|f^{\prime }(b)|]}{8}.$$

(4)

Theorem 2 [[5], Theorem 1]

If f is differentiable on $[a,b]$ such that $|f^{\prime }(x){|}^q$ is a convex function on $[a,b]$ for $q\ge 1$, then

$$|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x|\le \frac{b-a}{4}{\left[\frac{{|f^{\prime }(a)|}^q+{|f^{\prime }(b)|}^q}{2}\right]}^{1/q}.$$

(5)

Theorem 3 [[6], Theorem 2.3]

Let $f:I\to \mathbb{R}$ be differentiable on $I^{\circ }$, $a,b\in I^{\circ }$ with $a<b$, and $p>1$. If $|f^{\prime }(x){|}^{p/(p-1)}$ is convex on $[a,b]$, then

$$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x|\\ \le \frac{b-a}{16}{\left(\frac{4}{p+1}\right)}^{1/p}\{{\left[\right|f^{\prime }(a){|}^{p/(p-1)}+3|f^{\prime }(b){|}^{p/(p-1)}]}^{1-1/p}\\ +{\left[3\right|f^{\prime }(a){|}^{p/(p-1)}+|f^{\prime }(b){|}^{p/(p-1)}]}^{1-1/p}\}.\end{array}$$

(6)

Theorem 4 [[2], Theorem 2.1]

Let $A\subseteq \mathbb{R}$ be an open invex set with respect to $\eta :A\times A\to \mathbb{R}$ and $f:A\to \mathbb{R}$ be a differentiable function. If $|f^{\prime }(x)|$ is preinvex on A, then for every $a,b\in A$ with $\eta (a,b)\ne 0$

$$|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x|\le \frac{|\eta (a,b)|}{8}\left[\right|f^{\prime }(a)|+|f^{\prime }(b)\left|\right].$$

(7)

Theorem 5 [[2], Theorem 4.1]

Let $A\subseteq \mathbb{R}$ be an open invex set with respect to $\eta :A\times A\to \mathbb{R}$ and $\eta (a,b)\ne 0$ for all $a\ne b$. Suppose that $f:A\to \mathbb{R}$ is a twice differentiable function on A. If $|f^{″}(x)|$ is preinvex on A and $f^{″}$ is integrable on the η-path $P_{bc}$ for $c=b+\eta (a,b)$, then

$$|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x|\le \frac{{[\eta (a,b)]}^2}{24}\left[\right|f^{″}(a)|+|f^{″}(b)\left|\right].$$

(8)

Theorem 6 [[2], Theorem 4.3]

Let $A\subseteq \mathbb{R}$ be an open invex set with respect to $\eta :A\times A\to \mathbb{R}$ and $\eta (a,b)\ne 0$ for all $a\ne b$. Suppose that $f:A\to \mathbb{R}$ is a twice differentiable function on A and $|f^{″}(x)|$ is preinvex on A. If $q>1$ and $f^{″}$ is integrable on the η-path $P_{bc}$ for $c=b+\eta (a,b)$, then

$$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x|\\ \le \frac{{[\eta (a,b)]}^2}{12}{\left(\frac{1}{2}\right)}^{1/q}{\left[\right|f^{″}(a){|}^q+|f^{″}(b){|}^q]}^{1/q}.\end{array}$$

(9)

Recently, some related inequalities for preinvex functions were also obtained in [7, 8]. Some integral inequalities of Hermite-Hadamard type for other kinds of convex functions were also established in [9–16] and references cited therein.

In this paper, by creating an integral identity involving an n-times differentiable function, the authors will establish some new Hermite-Hadamard type inequalities for preinvex functions and generalize some of the above mentioned results.

2 A lemma

In order to obtain our main results, we need the following lemma.

Lemma 1 For $n\in \mathbb{N}$, let $A\subseteq \mathbb{R}$ be an open invex set with respect to $\eta :A\times A\to \mathbb{R}$ and let $a,b\in A$ with $\eta (a,b)\ne 0$ for all $a\ne b$. If $f:A\to \mathbb{R}$ is an n-times differentiable function on A and $f^{(n)}$ is integrable on the η-path $P_{bc}$ for $c=b+\eta (a,b)$, then

$$\begin{array}{r}\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{4[(k+1)!]}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]\\ =\frac{{[\eta (a,b)]}^n}{4(n!)}{\int }_0^1[{(1-t)}^{n-1}(2t+n-2)+{(-t)}^{n-1}(2t-n)]f^{(n)}(b+t\eta (a,b))\mathrm{d}t,\end{array}$$

(10)

where the above summation is zero for $n=1$.

Proof Since $a,b\in A$ and A is an invex set with respect to η, for every $t\in [0,1]$, we have $b+t\eta (a,b)\in A$. When $n=1$, integrating by parts in the right-hand side of (1) gives

$$\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x=\frac{\eta (a,b)}{2}{\int }_0^1(2t-1)f^{\prime }(b+t\eta (a,b))\mathrm{d}t.$$

Hence, the identity (1) holds for $n=1$.

When $n=m-1$ and $m\ge 2$, suppose that the identity (1) is valid.

When $n=m$, by the hypothesis, we have

$$\begin{array}{r}\frac{{[\eta (a,b)]}^m}{4(m!)}{\int }_0^1[{(1-t)}^{m-1}(2t+m-2)+{(-t)}^{m-1}(2t-m)]f^{(m)}(b+t\eta (a,b))\mathrm{d}t\\ =\frac{{[\eta (a,b)]}^{m-1}}{4(m!)}\{{(-1)}^{m-1}(2-m)f^{(m-1)}(b+\eta (a,b))-(m-2)f^{(m-1)}(b)\\ -m{\int }_0^1[{(1-t)}^{m-2}(3-2t-m)+{(-t)}^{m-2}(m-1-2t)]f^{(m-1)}(b+t\eta (a,b))\mathrm{d}t\}\\ =\frac{{[\eta (a,b)]}^{m-1}(2-m)}{4(m!)}[f^{(m-1)}(b)+{(-1)}^{m-1}{f}^{(m-1)}(b+\eta (a,b))]\\ +\frac{{[\eta (a,b)]}^{m-1}}{4[(m-1)!]}{\int }_0^1[{(1-t)}^{m-2}(2t+m-3)\\ +{(-t)}^{m-2}(2t-m+1)]f^{(m-1)}(b+t\eta (a,b))\mathrm{d}t\\ =\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\sum _{k=1}^{m-1}\frac{{[\eta (a,b)]}^k(1-k)}{4[(k+1)]!}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))].\end{array}$$

Therefore, when $n=m$, the identity (1) holds. By induction, the proof of Lemma 1 is complete. □

Remark 1 When $n=1$ and $n=2$ in (1), respectively, we obtain the identities

$$\begin{array}{r}\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ =\frac{\eta (a,b)}{2}{\int }_0^1(2t-1)f^{\prime }(b+t\eta (a,b))\mathrm{d}t\end{array}$$

and

$$\begin{array}{r}\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ =\frac{{[\eta (a,b)]}^2}{2}{\int }_0^{1}t(1-t)f^{″}(b+t\eta (a,b))\mathrm{d}t,\end{array}$$

which may be found in [2].

3 Hermite-Hadamard type inequalities for preinvex functions

Now we start out to establish some new Hermite-Hadamard type inequalities for n-times differentiable and preinvex functions.

Theorem 7 For $n\in \mathbb{N}$ and $n\ge 2$, let $A\subseteq \mathbb{R}$ be an open invex set with respect to $\eta :A\times A\to \mathbb{R}$ and $a,b\in A$ with $\eta (a,b)\ne 0$ for all $a\ne b$. Suppose that $f:A\to \mathbb{R}$ is an n-times differentiable function on A and $f^{(n)}$ is integrable on the η-path $P_{bc}$ for $c=b+\eta (a,b)$. If $|f^{(n)}{|}^q$ is preinvex on A for $q\ge 1$, then

$$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{4[(k+1)!]}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]|\\ \le \frac{|\eta (a,b){|}^n{(n-1)}^{1-1/q}}{4[(n+1)!]{(n+2)}^{1/q}}\{{\left[n\right|f^{(n)}(a){|}^q+(n^2-2)|f^{(n)}(b){|}^q]}^{1/q}\\ +{\left[(n^2-2)\right|f^{(n)}(a){|}^q+n|f^{(n)}(b){|}^q]}^{1/q}\}.\end{array}$$

(11)

Proof Since $a,b\in A$ and A is an invex set with respect to η, for every $t\in [0,1]$, we have $b+t\eta (a,b)\in A$. Using Lemma 1 and Hölder’s inequality yields

$$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{4[(k+1)!]}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]|\\ \le \frac{|\eta (a,b){|}^n}{4(n!)}[{\int }_0^1{(1-t)}^{n-1}(2t+n-2)|f^{(n)}(b+t\eta (a,b))|\mathrm{d}t\\ +{\int }_0^1{t}^{n-1}(n-2t)\left|f^{(n)}(b+t\eta (a,b))\right|\mathrm{d}t]\\ \le \frac{|\eta (a,b){|}^n}{4(n!)}\{{[{\int }_0^1{(1-t)}^{n-1}(2t+n-2)\mathrm{d}t]}^{1-1/q}\\ \times {[{\int }_0^1{(1-t)}^{n-1}(2t+n-2)\left(t\right|f^{(n)}(a){|}^q+(1-t)|f^{(n)}(b){|}^q)\mathrm{d}t]}^{1/q}\\ +{[{\int }_0^1{t}^{n-1}(n-2t)\mathrm{d}t]}^{1-1/q}[{\int }_0^1{t}^{n-1}(n-2t)(t|f^{(n)}(a){|}^q\\ {+(1-t)|f^{(n)}(b){|}^q)\mathrm{d}t]}^{1/q}\}\\ =\frac{|\eta (a,b){|}^n{(n-1)}^{1-1/q}}{4[(n+1)!]{(n+2)}^{1/q}}\{{\left[n\right|f^{(n)}(a){|}^q+(n^2-2)|f^{(n)}(b){|}^q]}^{1/q}\\ +{\left[(n^2-2)\right|f^{(n)}(a){|}^q+n|f^{(n)}(b){|}^q]}^{1/q}\}.\end{array}$$

Theorem 7 is thus proved. □

Corollary 1 Under the assumptions of Theorem  7,

• 1. if $q=1$, then

  $$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{4[(k+1)!]}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]|\\ \le \frac{(n-1)|\eta (a,b){|}^n}{4[(n+1)!]}\left[\right|f^{(n)}(a)|+|f^{(n)}(b)\left|\right];\end{array}$$

• 2. if $q=1$ and $n=2$, then the inequality (8) is valid.

Theorem 8 For $n\in \mathbb{N}$ and $n\ge 2$, let $A\subseteq \mathbb{R}$ be an open invex set with respect to $\eta :A\times A\to \mathbb{R}$ and $a,b\in A$ with $\eta (a,b)\ne 0$ for all $a\ne b$. Suppose that $f:A\to \mathbb{R}$ is an n-times differentiable function on A and $f^{(n)}$ is integrable on the η-path $P_{bc}$ for $c=b+\eta (a,b)$. If $|f^{(n)}{|}^q$ is preinvex on A for $q>1$, then

$$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\frac{1}{4}\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{(k+1)!}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]|\\ \le \frac{|\eta (a,b){|}^n}{16(n!){[(q+1)(q+2)]}^{1/q}}{\left[\frac{4(q-1)}{nq-1}\right]}^{1-1/q}\\ \times \left\{\right[({(n-2)}^{q+2}-(n-2q-4)n^{q+1})|f^{(n)}(a){|}^q\\ {+(n^{q+2}-(n+2q+2){(n-2)}^{q+1})|f^{(n)}(b){|}^q]}^{1/q}\\ +[(n^{q+2}-(n+2q+2){(n-2)}^{q+1})|f^{(n)}(a){|}^q\\ {+({(n-2)}^{q+2}-(n-2q-4)n^{q+1})|f^{(n)}(b){|}^q]}^{1/q}\}.\end{array}$$

(12)

Proof For every $t\in [0,1]$, we have $b+t\eta (a,b)\in A$. By Lemma 1 and Hölder’s inequality, it follows that

$$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{4[(k+1)!]}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]|\\ \le \frac{|\eta (a,b){|}^n}{4(n!)}[{\int }_0^1{(1-t)}^{n-1}(2t+n-2)|f^{(n)}(b+t\eta (a,b))|\mathrm{d}t\\ +{\int }_0^1{t}^{n-1}(n-2t)\left|f^{(n)}(b+t\eta (a,b))\right|\mathrm{d}t]\\ \le \frac{|\eta (a,b){|}^n}{4(n!)}\{{\left[{\int }_0^1{(1-t)}^{q(n-1)/(q-1)}\mathrm{d}t\right]}^{1-1/q}\\ \times {\left[{\int }_0^1{(2t+n-2)}^q\left(t\right|f^{(n)}(a){|}^q+(1-t)|f^{(n)}(b){|}^q)\mathrm{d}t\right]}^{1/q}\\ +{\left[{\int }_0^1{t}^{q(n-1)/(q-1)}\mathrm{d}t\right]}^{1-1/q}{\left[{\int }_0^1{(n-2t)}^q\left(t\right|f^{(n)}(a){|}^q+(1-t)|f^{(n)}(b){|}^q)\mathrm{d}t\right]}^{1/q}\}\\ =\frac{|\eta (a,b){|}^n}{16(n!){[(q+1)(q+2)]}^{1/q}}{\left[\frac{4(q-1)}{nq-1}\right]}^{1-1/q}\\ \times \left\{\right[({(n-2)}^{q+2}-(n-2q-4)n^{q+1})|f^{(n)}(a){|}^q\\ {+(n^{q+2}-(n+2q+2){(n-2)}^{q+1})|f^{(n)}(b){|}^q]}^{1/q}\\ +[(n^{q+2}-(n+2q+2){(n-2)}^{q+1})|f^{(n)}(a){|}^q\\ {+({(n-2)}^{q+2}-(n-2q-4)n^{q+1})|f^{(n)}(b){|}^q]}^{1/q}\}.\end{array}$$

Theorem 8 is thus proved. □

Theorem 9 For $n\in \mathbb{N}$ and $n\ge 2$, let $A\subseteq \mathbb{R}$ be an open invex set with respect to $\eta :A\times A\to \mathbb{R}$ and $a,b\in A$ with $\eta (a,b)\ne 0$ for all $a\ne b$. Suppose that $f:A\to \mathbb{R}$ is an n-times differentiable function on A and $f^{(n)}$ is integrable on the η-path $P_{bc}$ for $c=b+\eta (a,b)$. If $|f^{(n)}{|}^q$ is preinvex on A for $q>1$, then

$$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{4[(k+1)!]}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]|\\ \le \frac{|\eta (a,b){|}^n}{4(n!){[(nq-q+1)(nq-q+2)]}^{1/q}}\\ \times {\left\{\frac{(q-1)[n^{(2q-1)/(q-1)}-{(n-2)}^{(2q-1)/(q-1)}]}{2(2q-1)}\right\}}^{1-1/q}\\ \times \{{\left[\right|f^{(n)}(a){|}^q+(nq-q+1)|f^{(n)}(b){|}^q]}^{1/q}\\ +{[(nq-q+1)|f^{(n)}(a){|}^q+|f^{(n)}(b){|}^q]}^{1/q}\}.\end{array}$$

(13)

Proof Since $a,b\in A$ and A is an invex set with respect to η, for every $t\in [0,1]$, we have $b+t\eta (a,b)\in A$. Utilizing Lemma 1 and Hölder’s inequality results in

$$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{4[(k+1)!]}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]|\\ \le \frac{|\eta (a,b){|}^n}{4(n!)}[{\int }_0^1{(1-t)}^{n-1}(2t+n-2)|f^{(n)}(b+t\eta (a,b))|\mathrm{d}t\\ +{\int }_0^1{t}^{n-1}(n-2t)\left|f^{(n)}(b+t\eta (a,b))\right|\mathrm{d}t]\\ \le \frac{|\eta (a,b){|}^n}{4(n!)}\{{\left[{\int }_0^1{(2t+n-2)}^{q/(q-1)}\mathrm{d}t\right]}^{1-1/q}\\ \times {\left[{\int }_0^1{(1-t)}^{q(n-1)}\left(t\right|f^{(n)}(a){|}^q+(1-t)|f^{(n)}(b){|}^q)\mathrm{d}t\right]}^{1/q}\\ +{\left[{\int }_0^1{(n-2t)}^{q/(q-1)}\mathrm{d}t\right]}^{1-1/q}{\left[{\int }_0^1{t}^{q(n-1)}\left(t\right|f^{(n)}(a){|}^q+(1-t)|f^{(n)}(b){|}^q)\mathrm{d}t\right]}^{1/q}\}\\ =\frac{|\eta (a,b){|}^n}{4(n!){[(nq-q+1)(nq-q+2)]}^{1/q}}\\ \times {\left\{\frac{(q-1)[n^{(2q-1)/(q-1)}-{(n-2)}^{(2q-1)/(q-1)}]}{2(2q-1)}\right\}}^{1-1/q}\\ \times \{{\left[\right|f^{(n)}(a){|}^q+(nq-q+1)|f^{(n)}(b){|}^q]}^{1/q}\\ +{[(nq-q+1)|f^{(n)}(a){|}^q+|f^{(n)}(b){|}^q]}^{1/q}\}.\end{array}$$

The proof of Theorem 9 is complete. □

Theorem 10 For $n\in \mathbb{N}$ and $n\ge 2$, let $A\subseteq \mathbb{R}$ be an open invex set with respect to $\eta :A\times A\to \mathbb{R}$ and $a,b\in A$ with $\eta (a,b)\ne 0$ for all $a\ne b$. Suppose that $f:A\to \mathbb{R}$ is an n-times differentiable function on A and $f^{(n)}$ is integrable on the η-path $P_{bc}$ for $c=b+\eta (a,b)$. If $|f^{(n)}{|}^q$ is preinvex on A for $q>1$, then

$$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\frac{1}{4}\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{(k+1)!}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]|\\ \le \frac{|\eta (a,b){|}^n}{24(n!)}{\left[\frac{6(q-1)(nq-2)(n-1)}{(nq-1)(nq+q-2)}\right]}^{1-1/q}\\ \times \{{[(3n-2)|f^{(n)}(a){|}^q+(3n-4)|f^{(n)}(b){|}^q]}^{1/q}\\ +{[(3n-4)|f^{(n)}(a){|}^q+(3n-2)|f^{(n)}(b){|}^q]}^{1/q}\}.\end{array}$$

(14)

Proof Since $a,b\in A$ and A is an invex set with respect to η, for every $t\in [0,1]$, we have $b+t\eta (a,b)\in A$. Employing Lemma 1 and Hölder’s inequality leads to

$$\begin{array}{r}|\frac{f(b)+f(b+\eta (a,b))}{2}-\frac{1}{\eta (a,b)}{\int }_b^{b+\eta (a,b)}f(x)\mathrm{d}x\\ +\sum _{k=1}^{n-1}\frac{{[\eta (a,b)]}^k(1-k)}{4[(k+1)!]}[f^{(k)}(b)+{(-1)}^k{f}^{(k)}(b+\eta (a,b))]|\\ \le \frac{|\eta (a,b){|}^n}{4(n!)}[{\int }_0^1{(1-t)}^{n-1}(2t+n-2)|f^{(n)}(b+t\eta (a,b))|\mathrm{d}t\\ +{\int }_0^1{t}^{n-1}(n-2t)\left|f^{(n)}(b+t\eta (a,b))\right|\mathrm{d}t]\\ \le \frac{|\eta (a,b){|}^n}{4(n!)}\{{[{\int }_0^1{(1-t)}^{q(n-1)/(q-1)}(2t+n-2)\mathrm{d}t]}^{1-1/q}\\ \times {[{\int }_0^1(2t+n-2)\left(t\right|f^{(n)}(a){|}^q+(1-t)|f^{(n)}(b){|}^q)\mathrm{d}t]}^{1/q}\\ +{[{\int }_0^1{t}^{q(n-1)/(q-1)}(n-2t)\mathrm{d}t]}^{1-1/q}[{\int }_0^1(n-2t)(t|f^{(n)}(a){|}^q\\ {+(1-t)|f^{(n)}(b){|}^q)\mathrm{d}t]}^{1/q}\}\\ =\frac{|\eta (a,b){|}^n}{24(n!)}{\left[\frac{6(q-1)(nq-2)(n-1)}{(nq-1)(nq+q-2)}\right]}^{1-1/q}\left\{\right[(3n-2)|f^{(n)}(a){|}^q\\ {+(3n-4)|f^{(n)}(b){|}^q]}^{1/q}+{[(3n-4)|f^{(n)}(a){|}^q+(3n-2)|f^{(n)}(b){|}^q]}^{1/q}\}.\end{array}$$

The proof of Theorem 10 is complete. □