Estimates of upper bound for a k th order differentiable functions involving Riemann–Liouville integrals via higher order strongly h -preinvex functions

Shanhe Wu; Muhammad Uzair Awan; Marcela V. Mihai; Muhammad Aslam Noor; Sadia Talib

doi:10.1186/s13660-019-2146-z

Research
Open Access
Published: 28 August 2019

Estimates of upper bound for a kth order differentiable functions involving Riemann–Liouville integrals via higher order strongly h-preinvex functions

Shanhe Wu ORCID: orcid.org/0000-0002-7413-6563¹,
Muhammad Uzair Awan²,
Marcela V. Mihai³,
Muhammad Aslam Noor⁴ &
Sadia Talib²

Journal of Inequalities and Applications volume 2019, Article number: 227 (2019) Cite this article

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Abstract

In this paper the notion of higher order strongly h-preinvex functions is presented, which unifies several known classes of preinvexity. An identity related to the kth order differentiable functions and Riemann–Liouville integrals is established. The identity is then used to obtain some estimates of upper bound for the kth order differentiable functions involving Riemann–Liouville integrals via higher order strongly h-preinvex functions.

Introduction

We start by briefly summarizing the concepts on generalized convex functions which are related to the contents of this paper.

A set $K\subset \mathbb{R}$ is said to be convex if

$$\begin{aligned} (1-t)x+ty\in K,\quad \forall x,y\in K, t\in [0,1]. \end{aligned}$$

A function $f:K\to \mathbb{R}$ is said to be convex if

$$\begin{aligned} f \bigl((1-t)x+ty \bigr)\leq (1-t)f(x)+tf(y),\quad \forall x,y\in K, t\in [0,1]. \end{aligned}$$

For decades, the convexity properties of sets and functions have played the vital role in modern analysis, and they are particularly associated with the theory of inequalities. Applications of convexity in different fields of pure and applied sciences such as optimization theory, numerical analysis, economics etc. make this research topic more fascinating. Numerous generalizations of convexity have been proposed in the literature. For details, see [7]. Hanson [13] gave the notion of differentiable invex functions in connection with their special global optimum behavior. Craven [6] introduced the term “invex” for calling this class of functions due to their property described as “invariance by convexity”. Invex sets were first defined by Mititelu [19].

Let $K_{\eta }$ be a non-empty set in $\mathbb{R}$, and suppose that $\eta :\mathbb{R}^{n}\times \mathbb{R}^{n}\to \mathbb{R}^{n}$ is a continuous bifunction.

Definition 1.1

([19])

A set $K_{\eta }$ is said to be invex if

$$\begin{aligned} x+t\eta (y,x)\in K_{\eta },\quad \forall x,y\in K_{\eta },t\in [0,1]. \end{aligned}$$

Note that the invexity property of sets reduces to convexity if we take $\eta (y,x)=y-x$. Thus, every convex set is also an invex set with respect to $\eta (y,x)=y-x$, but the converse is not necessarily true.

Weir and Mond [31] introduced the notion of preinvex functions as follows.

Definition 1.2

([31])

A function $f:K_{\eta }\to \mathbb{R}$ is said to be preinvex if

$$\begin{aligned} f \bigl(x+t\eta (y,x) \bigr)\leq (1-t)f(x)+tf(y),\quad \forall x,y\in K_{\eta },t \in [0,1]. \end{aligned}$$

For recent studies on preinvex functions, interested readers are referred to [4, 13, 19, 20, 22,23,24,25].

Varosanec [30] introduced the class of h-convex functions. The idea at the time of introduction was to unify some generalizations of classical convexity, such as Breckner type of s-convex functions [5], Q-functions [12], and P-functions [11]. Now we know that this class also generalizes some other classes of classical convex functions. For details, see [9]. h-convex functions have received special attention by many researchers and, consequently, a considerable amount of research papers have been uniquely devoted to the study of this class. Noor et al. [25] extended this concept using the invexity property of sets and defined the notion of h-preinvex functions. They have observed that it contains several new and known classes of convexity.

Mohan and Neogy [20] discussed a very famous condition C.

Condition C

Let $\eta :\mathbb{R}^{n}\times \mathbb{R}^{n} \to \mathbb{R}^{n}$, we say that the bifunction $\eta (\cdot,\cdot)$ satisfies condition C if, for any $x,y\in \mathbb{R}^{n}$,

(1)
$\eta (x,x+t\eta (y,x))=-t\eta (y,x)$
(2)
$\eta (y,x+t\eta (y,x))=(1-t)\eta (y,x)$

for all $t\in [0,1]$.

Note that, for any $x,y\in \mathbb{R}^{n}$, $t_{1},t_{2}\in [0,1]$ and from condition C we can deduce

$$ \eta \bigl(x+t_{2}\eta (y,x),x+t_{1}\eta (y,x) \bigr)=(t_{2}-t_{1})\eta (y,x). $$

Karamardian [14] and Polyak [26] independently introduced strongly convex functions. Strong convexity is just a strengthening property of convexity.

Definition 1.3

A function $f:X\subset \mathbb{R}^{n}\to \mathbb{R}$ is said to be a strongly convex function with modulus $\mu >0$ if

$$\begin{aligned} f \bigl((1-t)x+ty \bigr)\leq (1-t)f(x)+tf(y)-\mu t(1-t) \Vert y-x \Vert ^{2},\quad \forall x,y\in X,t\in [0,1]. \end{aligned}$$

Karamardian [14] noticed that every differentiable function is strongly convex if and only if its gradient map is strongly monotone. Lin et al. [16] introduced higher order strongly convex functions to simplify the study of mathematical programs with equilibrium constraints. For some investigations on strongly convex functions, see [1,2,3, 17, 18, 21, 24, 26].

Definition 1.4

([16])

A function $f:X\subset \mathbb{R}^{n}\to \mathbb{R}$ is said to be a strongly convex function with order $\sigma >0$ and modulus $\mu >0$ if

$$\begin{aligned} f \bigl((1-t)x+ty \bigr)\leq (1-t)f(x)+tf(y)-\mu t(1-t) \Vert y-x \Vert ^{\sigma },\quad \forall x,y\in X,t\in [0,1]. \end{aligned}$$

Observe that when $\sigma =2$, then the above definition reduces to strong convexity in the classical sense. Lin et al. [16] have also shown that the higher order strong convexity of a function is equivalent to higher order strong monotonicity of the gradient map of the function.

Theory of convexity has also a great impact on theory of inequalities. One cannot deny the important role of convexity in the development of inequality theory. Many famous results in inequalities are due to the convexity property of functions. Hermite–Hadamard’s inequality, which is very simple in nature yet powerful, is one of the most extensively studied results for convex functions. This result provides us a necessary and sufficient condition for a function to be convex. It reads as follows.

Theorem 1.5

Let $f:I=[a,b]\subset \mathbb{R}\to \mathbb{R}$ be a convex function, then

$$\begin{aligned} f \biggl(\frac{a+b}{2} \biggr)\leq \frac{1}{b-a} \int _{a}^{b}f(x) \,\mathrm{d}x\leq \frac{f(a)+f(b)}{2}. \end{aligned}$$

For more and interesting details on Hermite–Hadamard’s inequalities, its generalizations and applications, see [8, 10].

Sarikaya et al. [27] utilized the concepts of fractional calculus and obtained a new variant of Hermite–Hadamard’s inequality via convex functions. In that paper authors also obtained a new fractional integral identity. This particular paper of Sarikaya et al. [27] has opened a new dimension of research. Since then many researchers have extensively utilized the concepts of fractional calculus and obtained numerous new and novel refinements of inequalities via convex functions and their generalizations. For more information, see [27,28,29, 32].

The aim of this article is to introduce the notion of higher order strongly h-preinvex functions. We also show that the class of higher order strongly h-preinvex functions unifies several other classes of strong preinvexity. We present an integral identity related to the kth order differentiable functions. Then, using this auxiliary result, we establish our main results that are some estimates of upper bound for kth order differentiable function via higher order strongly h-preinvex functions.

Higher order strongly h-preinvex functions

In this section, we introduce the higher order strongly h-preinvex functions and associated notions.

Definition 2.1

Let $h:(0,1)\to \mathbb{R}$ be a real function. A function $f:X\subset \mathbb{R}^{n}\to \mathbb{R}$ is said to be a higher order strongly h-preinvex function with order $\sigma >0$ and modulus $\mu \geq 0$ if

$$\begin{aligned} f \bigl(x+t\eta (y,x) \bigr)\leq h(1-t)f(x)+h(t)f(y)-\mu t(1-t) \bigl\Vert \eta (y,x) \bigr\Vert ^{ \sigma },\quad \forall x,y\in X,t\in [0,1]. \end{aligned}$$

Higher order strongly preinvex function is defined as follows.

Definition 2.2

([2])

A function $f:X\subset \mathbb{R}^{n}\to \mathbb{R}$ is said to be a higher order strongly preinvex function with order $\sigma >0$ and modulus $\mu \geq 0$ if

$$\begin{aligned} f \bigl(x+t\eta (y,x) \bigr)\leq (1-t)f(x)+tf(y)-\mu t(1-t) \bigl\Vert \eta (y,x) \bigr\Vert ^{\sigma },\quad \forall x,y\in X,t\in [0,1]. \end{aligned}$$

Higher order strongly s-preinvex function of Breckner type is defined as follows.

Definition 2.3

A function $f:X\subset \mathbb{R}^{n}\to \mathbb{R}$ is said to be a higher order strongly s-preinvex function of Breckner type with order $\sigma >0$, modulus $\mu \geq 0$, and $s\in (0,1]$ if

$$\begin{aligned} f \bigl(x+t\eta (y,x) \bigr)\leq (1-t)^{s}f(x)+t^{s}f(y)- \mu t(1-t) \bigl\Vert \eta (y,x) \bigr\Vert ^{\sigma },\quad \forall x,y \in X,t\in [0,1]. \end{aligned}$$

Higher order strongly s-preinvex function of Godunova–Levin type is defined as follows.

Definition 2.4

A function $f:X\subset \mathbb{R}^{n}\to \mathbb{R}$ is said to be a higher order strongly s-preinvex function of Godunova–Levin type with order $\sigma >0$, modulus $\mu \geq 0$, and $s\in (0,1)$ if

$$\begin{aligned} f \bigl(x+t\eta (y,x) \bigr)\leq \frac{1}{(1-t)^{s}}f(x)+\frac{1}{t^{s}}f(y)- \mu t(1-t) \bigl\Vert \eta (y,x) \bigr\Vert ^{\sigma },\quad \forall x,y\in X,t\in (0,1). \end{aligned}$$

Higher order strongly Q-preinvex function is defined as follows.

Definition 2.5

A function $f:X\subset \mathbb{R}^{n}\to \mathbb{R}$ is said to be a higher order strongly Q-preinvex function with order $\sigma >0$ and modulus $\mu \geq 0$ if

$$\begin{aligned} f \bigl(x+t\eta (y,x) \bigr)\leq \frac{1}{1-t}f(x)+\frac{1}{t}f(y)- \mu t(1-t) \bigl\Vert \eta (y,x) \bigr\Vert ^{\sigma },\quad \forall x,y \in X,t\in (0,1). \end{aligned}$$

Higher order strongly P-preinvex function is defined as follows.

Definition 2.6

A function $f:X\subset \mathbb{R}^{n}\to \mathbb{R}$ is said to be higher order strongly P-preinvex function with order $\sigma >0$ and modulus $\mu \geq 0$ if

$$\begin{aligned} f \bigl(x+t\eta (y,x) \bigr)\leq f(x)+f(y)-\mu t(1-t) \bigl\Vert \eta (y,x) \bigr\Vert ^{\sigma }, \quad \forall x,y\in X,t\in (0,1). \end{aligned}$$

Remark 2.7

Note that if we take $h(t)=t,t^{s},t^{-s},t^{-1}$ and $h(t)=1$, then Definition 2.1 reduces to Definitions 2.2, 2.3, 2.4, 2.5 and Definition 2.6 respectively. Also it is obvious that if we take $\eta (y,x)=y-x$ in Definition 2.1, then we have the class of higher order strongly h-convex functions. It is worth to mention here that to the best of our knowledge this class is also new in the literature. Similarly, for different suitable choices of the function $h(\cdot)$, we have other classes of higher order strong convexity. Thus the class of higher order strongly h-preinvex functions is a quite unifying one.

Auxiliary result

In this section we shall prove an auxiliary result which plays an important role in dealing with subsequent results. Before we start to describe the auxiliary result, let us recall the classic definition of Riemann–Liouville fractional integrals.

Definition 3.1

([15])

Let ${f}\in L[c,d]$, where $c\geq 0$. The Riemann–Liouville integrals $J_{c+}^{\nu }{f}$ and $J_{d-}^{\nu }{f}$, of order $\nu >0$, are defined by

$$ J_{c+}^{\nu }{f}(x)=\frac{1}{\varGamma (\nu )} \int _{c}^{x}(x-t)^{\nu -1} {f}(t) \,\mathrm{d}t\quad \text{for }x>c $$

and

$$ J_{d-}^{\nu }{f}(x)=\frac{1}{\varGamma (\nu )} \int _{x}^{d}(t-x)^{\nu -1} {f}(t) \,\mathrm{d}t\quad \text{for }x< d, $$

respectively. Here, $\varGamma (\nu )=\int _{0}^{\infty }e^{-t}t^{\nu -1} \,\mathrm{d}t$ is the gamma function. We also make the convention

$$ J_{c+}^{0}{f}(x)=J_{d-}^{0}{f}(x)={f}(x). $$

Some of our calculations need beta and hypergeometric functions, which are respectively defined as

$$ \mathrm{B}(x,y)= \int _{0}^{1}t^{x-1}(1-t)^{y-1} \,\mathrm{d}t. $$

The integral form of the hypergeometric function is

$$ _{2}f_{1}(x,y;c;z)=\frac{1}{\mathrm{B}(y,c-y)} \int _{0}^{1}t^{y-1}(1-t)^{c-y-1}(1-zt)^{-x} \,\mathrm{d}t $$

for $|z|<1$, $c>y>0$.

Lemma 3.2

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable. If $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $0<\alpha$, and $n\in \mathbb{N}$, then

$$\begin{aligned} &\mathcal{H}(k,n,\alpha ,a,b) (f) \\ &\quad = \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[f^{(k)} \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr)+ f^{(k)} \biggl(a+\frac{n+t}{n+1} \eta (b,a) \biggr) \biggr] \,\mathrm{d}t, \end{aligned}$$

where

$$\begin{aligned} &\mathcal{H}(k,n,\alpha ,a,b) (f) \\ &\quad = \biggl(\frac{n+1}{{\eta (b,a)}} \biggr)^{k+\alpha }\varGamma (\alpha +k) \bigl[J_{(a+\frac{n}{n+1}\eta (b,a))^{+}}^{\alpha }f \bigl(a+\eta (b,a) \bigr)+(-1)^{k}J _{(a+\frac{1}{n+1}\eta (b,a))^{-}}^{\alpha }f(a) \bigr] \\ &\qquad{}-\sum_{j=1}^{k} \frac{\varGamma (\alpha +k)}{\varGamma (\alpha +k-j+1)} \biggl(\frac{n+1}{{\eta (b,a)}} \biggr)^{j} \\ & \qquad {}\times \biggl[f^{(k-j)} \biggl(a+\frac{n}{n+1}\eta (b,a) \biggr) +(-1)^{j}f ^{(k-j)} \biggl(a+ \frac{1}{n+1}\eta (b,a) \biggr) \biggr]. \end{aligned}$$

(3.1)

Proof

Let

$$\begin{aligned} & \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[f^{(k)} \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr)+ f^{(k)} \biggl(a+\frac{n+t}{n+1} \eta (b,a) \biggr) \biggr] \,\mathrm{d}t \\ &\quad = \int _{0}^{1}(1-t)^{\alpha +k-1}f^{(k)} \biggl(a+\frac{1-t}{n+1} \eta (b,a) \biggr)\,\mathrm{d}t\\&\qquad{} + \int _{0}^{1}(1-t)^{\alpha +k-1}f^{(k)} \biggl(a+\frac{n+t}{n+1}\eta (b,a) \biggr)\,\mathrm{d}t \\ &\quad =I_{1}+I_{2}. \end{aligned}$$

We first calculate $I_{1}$, integration by parts gives

$$\begin{aligned} I_{1} &= \int _{0}^{1}(1-t)^{\alpha +k-1}f^{(k)} \biggl(a+\frac{1-t}{n+1} \eta (b,a) \biggr)\,\mathrm{d}t \\ &= \biggl[-\frac{n+1}{\eta (b,a)}(1-t)^{\alpha +k-1}f^{(k-1)} \biggl(a+\frac{1-t}{n+1}\eta (b,a) \biggr) \biggr]\bigg\vert _{0} ^{1} \\ &\quad {}-\frac{(\alpha +k-1)(n+1)}{{\eta (b,a)}} \int _{0}^{1}(1-t)^{ \alpha +k-2}f^{(k-1)} \biggl(a+\frac{1-t}{n+1}\eta (b,a) \biggr) \,\mathrm{d}t. \end{aligned}$$

Using integration by parts again, we have

$$\begin{aligned} I_{1} &=\frac{n+1}{{\eta (b,a)}}f^{(k-1)} \biggl(a+ \frac{1}{n+1}\eta (b,a) \biggr)-\frac{( \alpha +k-1)(n+1)^{2}}{\eta ^{2}(b,a)}f^{(k-2)} \biggl(a+\frac{1}{n+1} \eta (b,a) \biggr) \\ &\quad {}+\frac{(\alpha +k-1)(\alpha +k-2)(n+1)^{2}}{\eta ^{2}(b,a)} \int _{0}^{1}(1-t)^{\alpha +k-3}f^{(k-2)} \biggl(a+\frac{1-t}{n+1}\eta (b,a) \biggr) \,\mathrm{d}t. \end{aligned}$$

Repeat the integration by parts successively and obtain, after k integrations,

$$\begin{aligned} I_{1} &=\sum_{j=1}^{k} \frac{(-1)^{j-1}}{\alpha +k} \biggl(\frac{n+1}{ {\eta (b,a)}} \biggr)^{j}f^{(k-j)} \biggl(a+\frac{1}{n+1}\eta (b,a) \biggr) \prod _{p=0}^{j-1}(\alpha +k-p) \\ &\quad {}+\frac{(-1)^{k}}{\alpha +k} \biggl(\frac{n+1}{{\eta (b,a)}} \biggr) ^{k} \Biggl[\prod_{p=0}^{k}( \alpha +k-p) \Biggr] \int _{0}^{1}(1-t)^{ \alpha -1}f \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr)\,\mathrm{d}t \\ &=\sum_{j=1}^{k}\frac{(-1)^{j-1}}{\alpha +k} \biggl(\frac{n+1}{{\eta (b,a)}} \biggr) ^{j}f^{(k-j)} \biggl(a+ \frac{1}{n+1}\eta (b,a) \biggr)\prod_{p=0}^{j-1}( \alpha +k-p) \\ &\quad {}+(-1)^{k}\frac{\varGamma (\alpha +k)}{\varGamma (\alpha )} \biggl( \frac{n+1}{ {\eta (b,a)}} \biggr)^{k} \int _{0}^{1}(1-t)^{\alpha -1}f \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr)\,\mathrm{d}t \\ &=\sum_{j=1}^{k}(-1)^{j-1} \biggl(\frac{n+1}{{\eta (b,a)}} \biggr) ^{j}\frac{\varGamma (\alpha +k)}{\varGamma (\alpha +k-j+1)}f^{(k-j)} \biggl(a+ \frac{1}{n+1}\eta (b,a) \biggr) \\ &\quad {}+(-1)^{k}\varGamma (\alpha +k) \biggl(\frac{n+1}{{\eta (b,a)}} \biggr) ^{\alpha +k}J_{(a+\frac{1}{n+1}\eta (b,a))^{-}}^{\alpha }f(a). \end{aligned}$$

A similar method as above is used to compute $I_{2}$, we get

$$\begin{aligned} I_{2} &= \int _{0}^{1}(1-t)^{\alpha +k-1}f^{(k)} \biggl(a+\frac{n+t}{n+1} \eta (b,a) \biggr)\,\mathrm{d}t \\ &=-\frac{n+1}{{\eta (b,a)}}f^{(k-1)} \biggl(a+\frac{n}{n+1}\eta (b,a) \biggr)-( \alpha +k-1) \biggl(\frac{n+1}{{\eta (b,a)}} \biggr)^{2}f^{(k-2)} \biggl(a+ \frac{n}{n+1}\eta (b,a) \biggr) \\ &\quad {}-(\alpha +k-1) (\alpha +k-2) \biggl(\frac{n+1}{{\eta (b,a)}} \biggr) ^{3}f^{(k-3)} \biggl(a+\frac{n}{n+1}\eta (b,a) \biggr) \\ &\quad {}+(\alpha +k-1) (\alpha +k-2) (\alpha +k-3) \biggl(\frac{n+1}{ {\eta (b,a)}} \biggr)^{3}\\ &\quad {}\times \int _{0}^{1}(1-t)^{\alpha +k-4}f^{(k-3)} \biggl(a+\frac{n+t}{n+1}\eta (b,a) \biggr)\,\mathrm{d}t \\ &\vdots \\ &=-\sum_{j=1}^{k}\frac{\prod_{p=0}^{j-1}(\alpha +k-p)}{\alpha +k} \biggl(\frac{n+1}{{\eta (b,a)}} \biggr)^{j}f^{(k-j)} \biggl(a+ \frac{n}{n+1}\eta (b,a) \biggr) \\ &\quad {}+\frac{\prod_{p=0}^{k}(\alpha +k-p)}{\alpha +k} \biggl(\frac{n+1}{ {\eta (b,a)}} \biggr)^{k} \int _{0}^{1}(1-t)^{\alpha -1}f \biggl(a+ \frac{n+t}{n+1}\eta (b,a) \biggr)\,\mathrm{d}t \\ &=-\sum_{j=1}^{k} \biggl( \frac{n+1}{{\eta (b,a)}} \biggr)^{j}\frac{ \varGamma (\alpha +k)}{\varGamma (\alpha +k-j+1)}f^{(k-j)} \biggl(a+ \frac{n}{n+1}\eta (b,a) \biggr) \\ &\quad {}+\frac{\varGamma (\alpha +k)}{\varGamma (\alpha )} \biggl(\frac{n+1}{ {\eta (b,a)}} \biggr)^{k} \int _{0}^{1}(1-t)^{\alpha -1}f \biggl(a+ \frac{n+t}{n+1}\eta (b,a) \biggr)\,\mathrm{d}t \\ &=-\sum_{j=1}^{k} \biggl( \frac{n+1}{{\eta (b,a)}} \biggr)^{j}\frac{ \varGamma (\alpha +k)}{\varGamma (\alpha +k-j+1)}f^{(k-j)} \biggl(a+ \frac{n}{n+1}\eta (b,a) \biggr) \\ & \quad {}+\varGamma (\alpha +k) \biggl(\frac{n+1}{{\eta (b,a)}} \biggr)^{\alpha +k}J _{(a+\frac{n}{n+1}\eta (b,a))^{+}}^{\alpha }f \bigl(a+\eta (b,a) \bigr). \end{aligned}$$

Now, by summing up $I_{1}$ and $I_{2}$, we have

$$\begin{aligned} I_{1} +I_{2} & =\sum_{j=1}^{k} \frac{\varGamma (\alpha +k)}{\varGamma (\alpha +k-j+1)} \biggl(\frac{n+1}{{\eta (b,a)}} \biggr)^{j} \biggl[(-1)^{j-1}f^{(k-j)} \biggl(a+\frac{1}{n+1}\eta (b,a) \biggr) \\ &\quad {}-f^{(k-j)} \biggl(a+\frac{n}{n+1}\eta (b,a) \biggr) \biggr] \\ &\quad {}+\varGamma (\alpha +k) \biggl(\frac{n+1}{{\eta (b,a)}} \biggr) ^{\alpha +k} \bigl[J_{(a+\frac{n}{n+1}\eta (b,a))^{+}}^{\alpha }f \bigl(a+ \eta (b,a) \bigr) +(-1)^{k}J_{(a+\frac{1}{n+1}\eta (b,a))^{-}}^{\alpha }f(a) \bigr]. \end{aligned}$$

After convenient arrangement, we deduce the identity described in Lemma 3.2. □

Main results

We consider some estimates of upper bound for the function $\mathcal{H}(k,n,\alpha ,a,b)(f)$ via higher order strongly h-preinvex functions, our main results are stated in the following theorems.

Theorem 4.1

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $0<\alpha$, and $n\in \mathbb{N}$. If $|f^{(k)}|$ is a higher order strongly h-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \phi (\alpha ,k,n,t) \bigl[ \bigl\vert f^{(k)}(a) \bigr\vert + \bigl\vert f^{(k)} (b) \bigr\vert \bigr]\\ &\qquad {}-\frac{2\mu}{(n+1)^{2}}\left(\frac{n(\alpha+k+2)+1}{(\alpha+k+1)(\alpha+k+2)}\right) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma }, \end{aligned}$$

where

$$\begin{aligned} \phi (\alpha ,k,n,t):= \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[h \biggl(\frac{n+t}{n+1} \biggr)+h \biggl(\frac{1-t}{n+1} \biggr) \biggr]\,\mathrm{d}t. \end{aligned}$$

(4.1)

Proof

Utilizing Lemma 3.2, the property of the modulus and the fact that $|f^{(k)}|$ is a higher order strongly h-preinvex function, we have

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad = \biggl\vert \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[f^{(k)} \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr)+ f^{(k)} \biggl(a+\frac{n+t}{n+1} \eta (b,a) \biggr) \biggr] \,\mathrm{d}t \biggr\vert \\ &\quad \leq \biggl\vert \int _{0}^{1}(1-t)^{\alpha +k-1}f^{(k)} \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr) \,\mathrm{d}t \biggr\vert \\ &\qquad {}+ \biggl\vert \int _{0} ^{1}(1-t)^{\alpha +k-1}f^{(k)} \biggl(a+\frac{n+t}{n+1}\eta (b,a) \biggr) \,\mathrm{d}t \biggr\vert \\ &\quad \leq \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[h \biggl(\frac{n+t}{n+1} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert +h \biggl(\frac{1-t}{n+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert \\ &\qquad{} -\frac{\mu }{(n+1)^{2}}(n+t) (1-t) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr] \,\mathrm{d}t \\ &\qquad {}+ \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[h \biggl(\frac{1-t}{n+1} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert +h \biggl(\frac{n+t}{n+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert \\ & \qquad{} -\frac{\mu }{(n+1)^{2}}(n+t) (1-t) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr] \,\mathrm{d}t \\ &\quad =\phi (\alpha ,k,n,t) \bigl[ \bigl\vert f^{(k)}(a) \bigr\vert + \bigl\vert f^{(k)} (b) \bigr\vert \bigr]\\ &\qquad {}-\frac{2\mu}{(n+1)^{2}}\left(\frac{n(\alpha+k+2)+1}{(\alpha+k+1)(\alpha+k+2)}\right) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma }. \end{aligned}$$

This completes the proof of Theorem 4.1. □

We now discuss some special cases which can be deduced directly from Theorem 4.1.

I. If we take $h(t)=t$, then we have the result for higher order strongly preinvex functions.

Corollary 4.2

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, and $n\in \mathbb{N}$. If $|f^{(k)}|$ is a higher order strongly preinvex function, then

$$\begin{aligned} \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert &\leq \phi ^{*}(\alpha ,k,n) \bigl[ \bigl\vert f^{(k)}(a) \bigr\vert + \bigl\vert f^{(k)} (b) \bigr\vert \bigr] \\ & \quad {}-\frac{2\mu}{(n+1)^{2}}\left(\frac{n(\alpha+k+2)+1}{(\alpha+k+1)(\alpha+k+2)}\right) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma }, \end{aligned}$$

where

$$\begin{aligned} \phi ^{*}(\alpha ,k,n):=\frac{1}{\alpha +k}. \end{aligned}$$

II. If we put $h(t)=t^{s}$, then we have the result for Breckner type of higher order strongly s-preinvex functions.

Corollary 4.3

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, and $n\in \mathbb{N}$. If $|f^{(k)}|$ is Breckner type of a higher order strongly s-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \phi ^{**}(\alpha ,k,n) \bigl[ \bigl\vert f^{(k)}(a) \bigr\vert + \bigl\vert f^{(k)} (b) \bigr\vert \bigr]\\ &\qquad {}-\frac{2\mu}{(n+1)^{2}}\left(\frac{n(\alpha+k+2)+1}{(\alpha+k+1)(\alpha+k+2)}\right) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma }, \end{aligned}$$

where

$$\begin{aligned} \phi ^{**}(\alpha ,k,n) &:= \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[ \biggl(\frac{n+t}{n+1} \biggr) ^{s}+ \biggl( \frac{1-t}{n+1} \biggr)^{s} \biggr]\,\mathrm{d}t \\ &:=\frac{1}{(n+1)^{s}} \biggl[n^{s}B(1,\alpha +k) _{2}f_{1} \biggl(-s,1; \alpha +k+1;- \frac{1}{n} \biggr)+\frac{1}{\alpha +k+s} \biggr]. \end{aligned}$$

III. If we put $h(t)=t^{-s}$, then we have the result for Godunova–Levin type of higher order strongly s-preinvex functions.

Corollary 4.4

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, and $n\in \mathbb{N}$. If $|f^{(k)}|$ is Godunova–Levin type of a higher order strongly s-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \phi ^{***}(\alpha ,k,n) \bigl[ \bigl\vert f^{(k)}(a) \bigr\vert + \bigl\vert f^{(k)} (b) \bigr\vert \bigr]\\ &\qquad {}-\frac{2\mu}{(n+1)^{2}}\left(\frac{n(\alpha+k+2)+1}{(\alpha+k+1)(\alpha+k+2)}\right) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma }, \end{aligned}$$

where

$$\begin{aligned} \phi ^{***}(\alpha ,k,n) &:= \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[ \biggl(\frac{n+t}{n+1} \biggr) ^{-s}+ \biggl( \frac{1-t}{n+1} \biggr)^{-s} \biggr]\,\mathrm{d}t \\ &:=(n+1)^{s} \biggl[n^{-s}B(1,\alpha +k){}_{2}f_{1} \biggl(s,1;\alpha +k+1;- \frac{1}{n} \biggr)+\frac{1}{\alpha +k-s} \biggr]. \end{aligned}$$

IV. If we choose $h(t)=1$, then we have the result for higher order strongly P-preinvex functions.

Corollary 4.5

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, and $n\in \mathbb{N}$. If $|f^{(k)}|$ is a higher order strongly P-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \frac{2}{\alpha +k} \bigl[ \bigl\vert f^{(k)}(a) \bigr\vert + \bigl\vert f^{(k)} (b) \bigr\vert \bigr]-\frac{2\mu}{(n+1)^{2}}\left(\frac{n(\alpha+k+2)+1}{(\alpha+k+1)(\alpha+k+2)}\right) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma }. \end{aligned}$$

Theorem 4.6

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $0<\alpha$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ where $\frac{1}{p}+\frac{1}{q}=1$, $q>1$ is a higher order strongly h-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl(\frac{1}{p(\alpha +k-1)+1} \biggr)^{\frac{1}{p}} \biggl\{ \biggl[ \int _{0}^{1} \biggl(h \biggl( \frac{n+t}{n+1} \biggr) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}\\ &\qquad {}+h \biggl(\frac{1-t}{n+1} \biggr) \bigl\vert f ^{(k)} (b) \bigr\vert ^{q} \biggr) \,\mathrm{d}t \\ &\qquad {}-\frac{\mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \\ &\qquad {}+ \biggl[ \int _{0}^{1} \biggl(h \biggl( \frac{1-t}{n+1} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert ^{q}+h \biggl(\frac{n+t}{n+1} \biggr) \bigl\vert f^{(k)} \bigl(b \bigr) \bigr\vert ^{q} \biggr) \,\mathrm{d}t \\ &\qquad {}- \frac{\mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \biggr\} . \end{aligned}$$

Proof

Utilizing Lemma 3.2, Hölder’s inequality, and the fact that $|f^{(k)}|^{q}$ is a higher order strongly h-preinvex function, we have

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad = \biggl\vert \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[f^{(k)} \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr)+ f^{(k)} \biggl(a+\frac{n+t}{n+1} \eta (b,a) \biggr) \biggr] \,\mathrm{d}t \biggr\vert \\ &\quad \leq \biggl( \int _{0}^{1}(1-t)^{p(\alpha +k-1)}\,\mathrm{d}t \biggr) ^{\frac{1}{p}} \biggl( \int _{0}^{1} \biggl\vert f^{(k)} \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr) \biggr\vert ^{q} \,\mathrm{d}t \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1}(1-t)^{p(\alpha +k-1)}\,\mathrm{d}t \biggr) ^{\frac{1}{p}} \biggl( \int _{0}^{1} \biggl\vert f^{(k)} \biggl(a+ \frac{n+t}{n+1}\eta (b,a) \biggr) \biggr\vert ^{q} \,\mathrm{d}t \biggr)^{ \frac{1}{q}} \\ &\quad \leq \biggl(\frac{1}{p(\alpha +k-1)+1} \biggr)^{\frac{1}{p}} \biggl[ \int _{0}^{1} \biggl(h \biggl( \frac{n+t}{n+1} \biggr) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+h \biggl(\frac{1-t}{n+1} \biggr) \bigl\vert f ^{(k)}(b) \bigr\vert ^{q} \\ &\qquad {}-\frac{\mu }{(n+1)^{2}}(n+t) (1-t) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr) \,\mathrm{d}t \biggr]^{\frac{1}{q}} \\ &\qquad {}+ \biggl(\frac{1}{p(\alpha +k-1)+1} \biggr)^{\frac{1}{p}} \biggl[ \int _{0}^{1} \biggl(h \biggl( \frac{1-t}{n+1} \biggr) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+h \biggl(\frac{n+t}{n+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ &\qquad {}-\frac{\mu }{(n+1)^{2}}(n+t) (1-t) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr) \,\mathrm{d}t \biggr]^{\frac{1}{q}} \\ &\quad = \biggl(\frac{1}{p(\alpha +k-1)+1} \biggr)^{\frac{1}{p}} \biggl\{ \biggl[ \int _{0}^{1} \biggl(h \biggl( \frac{n+t}{n+1} \biggr) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}\\ &\qquad {}+h \biggl(\frac{1-t}{n+1} \biggr) \bigl\vert f ^{(k)}(b) \bigr\vert ^{q} \biggr) \,\mathrm{d}t \\ &\qquad {}- \frac{\mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \\ &\qquad {}+ \biggl[ \int _{0}^{1} \biggl(h \biggl( \frac{1-t}{n+1} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert ^{q}+h \biggl(\frac{n+t}{n+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \biggr) \,\mathrm{d}t \\ &\qquad {}- \frac{\mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \biggr\} . \end{aligned}$$

The proof of Theorem 4.6 is completed. □

In the following, we give four corollaries that follow from the special cases of Theorem 4.6.

I. If we choose $h(t)=t$, then we have the result for higher order strongly preinvex functions.

Corollary 4.7

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ where $\frac{1}{p}+\frac{1}{q}=1$, $q>1$ is a higher order strongly preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl(\frac{1}{p(\alpha +k-1)+1} \biggr)^{\frac{1}{p}} \biggl( \frac{1}{2(n+1)} \biggr) ^{\frac{1}{q}} \\ &\qquad {}\times \biggl\{ \biggl[(2n+1) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+ \bigl\vert f^{(k)} (b) \bigr\vert ^{q}- \frac{ \mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{\frac{1}{q}} \\ &\qquad {}+ \biggl[ \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+(2n+1) \bigl\vert f^{(k)} (b) \bigr\vert ^{q}- \frac{ \mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$

II. If we take $h(t)=t^{s}$, then we have the result for Breckner type of higher order strongly s-preinvex functions.

Corollary 4.8

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ where $\frac{1}{p}+\frac{1}{q}=1$, $q>1$ is Breckner type of a higher order strongly s-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl(\frac{1}{p(\alpha +k-1)+1} \biggr)^{\frac{1}{p}} \biggl( \frac{1}{(n+1)^{s}} \biggr) ^{\frac{1}{q}} \\ &\qquad {}\times \biggl\{ \biggl[ \biggl(\frac{(1+n)^{s+1}-n^{s+1}}{s+1} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert ^{q}+ \biggl( \frac{1}{s+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q}\\ &\qquad {}- \frac{\mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \\ &\qquad {}+ \biggl[ \biggl(\frac{1}{s+1} \biggr) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+ \biggl( \frac{(1+n)^{s+1}-n^{s+1}}{s+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q}\\ &\qquad {}- \frac{\mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \biggr\} . \end{aligned}$$

III. If we take $h(t)=t^{-s}$, then we have the result for Godunova–Levin type of higher order strongly s-preinvex functions.

Corollary 4.9

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ where $\frac{1}{p}+\frac{1}{q}=1$, $q>1$ is Godunova–Levin type of a higher order strongly s-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl(\frac{1}{p(\alpha +k-1)+1} \biggr)^{\frac{1}{p}}(n+1)^{ \frac{s}{q}} \\ &\qquad {}\times \biggl\{ \biggl[ \biggl(\frac{(1+n)^{1-s}-n^{1-s}}{1-s} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert ^{q}+ \biggl( \frac{1}{1-s} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q}\\ &\qquad {}- \frac{\mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \\ &\qquad {}+ \biggl[ \biggl(\frac{1}{1-s} \biggr) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+ \biggl( \frac{(1+n)^{1-s}-n^{1-s}}{1-s} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q}\\ &\qquad {}- \frac{\mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \biggr\} . \end{aligned}$$

IV. If we put $h(t)=1$, then we have the result for higher order strongly P-preinvex functions.

Corollary 4.10

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ where $\frac{1}{p}+\frac{1}{q}=1$, $q>1$ is a higher order strongly P-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq 2 \biggl(\frac{1}{p(\alpha +k-1)+1} \biggr)^{\frac{1}{p}} \biggl[ \bigl\vert f ^{(k)}(a) \bigr\vert ^{q}+ \bigl\vert f^{(k)}(b) \bigr\vert ^{q}- \frac{\mu (3n+1)}{6(n+1)^{2}} \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}}. \end{aligned}$$

Theorem 4.11

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $0<\alpha$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is a higher order strongly h-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl[\psi _{1}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{2}(\alpha ,k,n) \bigl\vert f ^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}} \\ &\qquad {}+ \biggl[\psi _{2}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{1}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}}, \end{aligned}$$

where

$$\begin{aligned} \psi _{1}(\alpha ,k,n):= \int _{0}^{1}(1-t)^{q(\alpha +k-1)}h \biggl( \frac{n+t}{n+1} \biggr) \,\mathrm{d}t \end{aligned}$$

(4.2)

and

$$\begin{aligned} \psi _{2}(\alpha ,k,n):= \int _{0}^{1}(1-t)^{q(\alpha +k-1)}h \biggl( \frac{1-t}{n+1} \biggr) \,\mathrm{d}t. \end{aligned}$$

(4.3)

Proof

Utilizing Lemma 3.2, Hölder’s inequality, and the fact that $|f^{(k)}|$ is a higher order strongly h-preinvex function, we have

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad = \biggl\vert \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[f^{(k)} \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr)+ f^{(k)} \biggl(a+\frac{n+t}{n+1} \eta (b,a) \biggr) \biggr] \,\mathrm{d}t \biggr\vert \\ &\quad \leq \biggl( \int _{0}^{1}1\,\mathrm{d}t \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1}(1-t)^{q(\alpha +k-1)} \biggl\vert f^{(k)} \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr) \biggr\vert ^{q}\,\mathrm{d}t \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1}1\,\mathrm{d}t \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1}(1-t)^{q(\alpha +k-1)} \biggl\vert f^{(k)} \biggl(a+ \frac{n+t}{n+1}\eta (b,a) \biggr) \biggr\vert ^{q}\,\mathrm{d}t \biggr)^{ \frac{1}{q}} \\ &\quad \leq \biggl[ \int _{0}^{1}(1-t)^{q(\alpha +k-1)} \biggl[h \biggl(\frac{n+t}{n+1} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert ^{q}+h \biggl(\frac{1-t}{n+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ &\qquad {}-\frac{\mu }{(n+1)^{2}}(n+t) (1-t) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]\,\mathrm{d}t \biggr]^{\frac{1}{q}} \\ &\qquad {}+ \biggl[ \int _{0}^{1}(1-t)^{q(\alpha +k-1)} \biggl[h \biggl(\frac{1-t}{n+1} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert ^{q}+h \biggl(\frac{n+t}{n+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ &\qquad {}-\frac{\mu }{(n+1)^{2}}(n+t) (1-t) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]\,\mathrm{d}t \biggr]^{\frac{1}{q}} \\ &\quad = \biggl[\psi _{1}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{2}(\alpha ,k,n) \bigl\vert f ^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}} \\ &\qquad {}+ \biggl[\psi _{2}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{1}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}}. \end{aligned}$$

This completes the proof of Theorem 4.11. □

We next discuss some special cases of Theorem 4.11.

I. If we put $h(t)=t$, then we have the result for higher order strongly preinvex functions.

Corollary 4.12

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is a higher order strongly preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl[\psi _{1}^{*}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{2}^{*}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}} \\ & \qquad {}+ \biggl[\psi _{2}^{*}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{1}^{*}(\alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}}, \end{aligned}$$

where

$$\begin{aligned} \psi _{1}^{*}(\alpha ,k,n):=\frac{1}{q(\alpha +k-1)+1}- \frac{1}{(n+1)[q( \alpha +k-1)+2]} \end{aligned}$$

and

$$\begin{aligned} \psi _{2}^{*}(\alpha ,k,n):=\frac{1}{(n+1)[q(\alpha +k-1)+2]}. \end{aligned}$$

II. If we take $h(t)=t^{s}$, then we have the result for Breckner type of higher order strongly s-preinvex functions.

Corollary 4.13

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is Breckner type of a higher order strongly s-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl[\psi _{1}^{**}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{2}^{**}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}} \\ & \qquad {}+ \biggl[\psi _{2}^{**}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{1}^{**}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}}, \end{aligned}$$

where

$$\begin{aligned} \psi _{1}^{**}(\alpha ,k,n) &:=\frac{1}{(n+1)^{s}} \int _{0}^{1}(1-t)^{q( \alpha +k-1)} (n+t )^{s}\,\mathrm{d}t \\ &:= \frac{n^{s}}{(n+1)^{s}} B \bigl(1,q(\alpha +k-1)+1 \bigr){}_{2}f_{1} \biggl(-s,1;q(\alpha +k-1)+2;- \frac{1}{n} \biggr) \end{aligned}$$

and

$$\begin{aligned} \psi _{2}^{**}(\alpha ,k,n):=\frac{1}{(n+1)^{s}}\cdot \frac{1}{q( \alpha +k-1)+s+1}. \end{aligned}$$

III. If we take $h(t)=t^{-s}$, then we have the result for Godunova–Levin type of higher order strongly s-preinvex functions.

Corollary 4.14

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is Godunova–Levin type of a higher order strongly s-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl[\psi _{1}^{***}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{2}^{***}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}} \\ & \qquad {}+ \biggl[\psi _{2}^{***}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\psi _{1}^{***}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}}, \end{aligned}$$

where

$$\begin{aligned} \psi _{1}^{***}(\alpha ,k,n) &:=(n+1)^{s} \int _{0}^{1}(1-t)^{q(\alpha +k-1)}(n+t)^{-s} \,\mathrm{d}t \\ &:= \frac{(n+1)^{s}}{n^{s}} B \bigl(1,q(\alpha +k-1)+1 \bigr){}_{2}f_{1} \biggl(s,1;q(\alpha +k-1)+2;- \frac{1}{n} \biggr) \end{aligned}$$

and

$$\begin{aligned} \psi _{2}^{***}(\alpha ,k,n):=\frac{(n+1)^{s}}{q(\alpha +k-1)-s+1}. \end{aligned}$$

IV. If we choose $h(t)=1$, then we have the result for higher order strongly P-preinvex functions.

Corollary 4.15

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is a higher order strongly P-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \frac{2}{(q(\alpha +k-1)+1)^{\frac{1}{q}}} \biggl[ \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+ \bigl\vert f ^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu (q(\alpha +k-1)+1) }{n+1} \biggl(\frac{1}{q(\alpha +k-1)+2}- \frac{1}{(n+1)(q( \alpha +k-1)+3)} \biggr) \parallel \eta (b,a) \parallel ^{\sigma} \biggr]^{\frac{1}{q}}. \end{aligned}$$

Theorem 4.16

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $0<\alpha$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is a higher order strongly h-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl(\frac{1}{\alpha +k} \biggr)^{1-\frac{1}{q}} \bigg\{ \biggl[ \theta _{1}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{2}(\alpha ,k,n) \bigl\vert f^{(k)}(b) \bigr\vert ^{q} \\ &\qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{ \sigma } \biggr]^{\frac{1}{q}} \\ &\qquad {}+ \biggl[\theta _{2}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{1}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q}\\ &\qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{\frac{1}{q}}\bigg\} , \end{aligned}$$

where

$$\begin{aligned} \theta _{1}(\alpha ,k,n):= \int _{0}^{1}(1-t)^{\alpha +k-1}h \biggl( \frac{n+t}{n+1} \biggr) \,\mathrm{d}t \end{aligned}$$

(4.4)

and

$$\begin{aligned} \theta _{2}(\alpha ,k,n):= \int _{0}^{1}(1-t)^{\alpha +k-1}h \biggl( \frac{1-t}{n+1} \biggr) \,\mathrm{d}t. \end{aligned}$$

(4.5)

Proof

Utilizing Lemma 3.2, power mean inequality, and the fact that $|f^{(k)}|$ is a higher order strongly h-preinvex function, we have

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad = \biggl\vert \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[f^{(k)} \biggl(a+ \frac{1-t}{n+1}\eta (b,a) \biggr)+ f^{(k)} \biggl(a+\frac{n+t}{n+1} \eta (b,a) \biggr) \biggr] \,\mathrm{d}t \biggr\vert \\ &\quad \leq \biggl( \int _{0}^{1}(1-t)^{\alpha +k-1}\,\mathrm{d}t \biggr) ^{1-\frac{1}{q}} \biggl( \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl\vert f^{(k)} \biggl(a+\frac{1-t}{n+1}\eta (b,a) \biggr) \biggr\vert ^{q}\,\mathrm{d}t \biggr) ^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1}(1-t)^{\alpha +k-1}\,\mathrm{d}t \biggr) ^{1-\frac{1}{q}} \biggl( \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl\vert f^{(k)} \biggl(a+\frac{n+t}{n+1}\eta (b,a) \biggr) \biggr\vert ^{q}\,\mathrm{d}t \biggr) ^{\frac{1}{q}} \\ &\quad \leq \biggl(\frac{1}{\alpha +k} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[h \biggl(\frac{n+t}{n+1} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert ^{q}\\ &\qquad {}+h \biggl(\frac{1-t}{n+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ &\qquad {}-\frac{\mu }{(n+1)^{2}}(n+t) (1-t) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]\,\mathrm{d}t \biggr]^{\frac{1}{q}} \\ &\qquad + \biggl[ \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl[h \biggl(\frac{1-t}{n+1} \biggr) \bigl\vert f ^{(k)}(a) \bigr\vert ^{q}+h \biggl(\frac{n+t}{n+1} \biggr) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ &\qquad {}-\frac{\mu }{(n+1)^{2}}(n+t) (1-t) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]\,\mathrm{d}t \biggr]^{\frac{1}{q}} \biggr\} \\ &\quad = \biggl(\frac{1}{\alpha +k} \biggr)^{1-\frac{1}{q}} \bigg\{ \biggl[ \theta _{1}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{2}(\alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ &\qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{ \sigma } \biggr]^{\frac{1}{q}} \\ &\qquad {}+ \biggl[\theta _{2}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{1}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q}\\ &\qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{\frac{1}{q}} \bigg\} . \end{aligned}$$

The proof of Theorem 4.16 is complete. □

In the following we discuss some special cases of Theorem 4.16.

I. If we put $h(t)=t$, then we have the result for higher order strongly preinvex functions.

Corollary 4.17

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is a higher order strongly preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl(\frac{1}{\alpha +k} \biggr)^{1-\frac{1}{q}} \bigg\{ \biggl[\theta _{1}^{*}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{2}^{*}(\alpha ,k,n) \bigl\vert f ^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \\ & \qquad {}+ \biggl[\theta _{2}^{*}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{1}^{*}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \bigg\} , \end{aligned}$$

where

$$\begin{aligned} \theta _{1}^{*}(\alpha ,k,n):=\frac{n(\alpha +k+1)+1}{(n+1)(\alpha +k)( \alpha +k+1)} \end{aligned}$$

and

$$\begin{aligned} \theta _{2}^{*}(\alpha ,k,n):=\frac{1}{(n+1)(\alpha +k+1)}. \end{aligned}$$

II. If we choose $h(t)=t^{s}$, then we have the result for Breckner type of higher order strongly s-preinvex functions.

Corollary 4.18

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is Breckner type of a higher order strongly s-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl(\frac{1}{\alpha +k} \biggr)^{1-\frac{1}{q}} \bigg\{ \biggl[\theta _{1}^{**}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{2}^{**}(\alpha ,k,n) \bigl\vert f ^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \\ & \qquad {}+ \biggl[\theta _{2}^{**}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{1}^{**}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \bigg\} , \end{aligned}$$

where

$$\begin{aligned} \theta _{1}^{**}(\alpha ,k,n) &:=\frac{1}{(n+1)^{s}} \int _{0}^{1}(1-t)^{ \alpha +k-1} (n+t )^{s}\,\mathrm{d}t, \\ &:= \frac{n^{s}}{(n+1)^{s}} B(1,\alpha +k){}_{2}f_{1} \biggl(-s,1;\alpha +k+1;-\frac{1}{n} \biggr) \end{aligned}$$

and

$$\begin{aligned} \theta _{2}^{**}(\alpha ,k,n):= \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl( \frac{1-t}{n+1} \biggr) ^{s}\,\mathrm{d}t=\frac{1}{(n+1)^{s}} \cdot \frac{1}{\alpha +k+s}. \end{aligned}$$

III. If we choose $h(t)=t^{-s}$, then we have the result for Godunova–Levin type of higher order strongly s-preinvex functions.

Corollary 4.19

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is Godunova–Levin type of a higher order strongly s-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \biggl(\frac{1}{\alpha +k} \biggr)^{1-\frac{1}{q}} \bigg\{ \biggl[\theta _{1}^{***}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{2}^{***}(\alpha ,k,n) \bigl\vert f ^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \\ & \qquad {}+ \biggl[\theta _{2}^{***}(\alpha ,k,n) \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+\theta _{1}^{***}( \alpha ,k,n) \bigl\vert f^{(k)} (b) \bigr\vert ^{q} \\ & \qquad {}-\frac{\mu }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}} \bigg\} , \end{aligned}$$

where

$$\begin{aligned} \theta _{1}^{***}(\alpha ,k,n) &:=(n+1)^{s} \int _{0}^{1}(1-t)^{\alpha +k-1} (n+t )^{-s}\,\mathrm{d}t, \\ &:= \frac{(n+1)^{s}}{n^{s}} B(1,\alpha +k){}_{2}f_{1} \biggl(s,1;\alpha +k+1;-\frac{1}{n} \biggr) \end{aligned}$$

and

$$\begin{aligned} \theta _{2}^{***}(\alpha ,k,n):= \int _{0}^{1}(1-t)^{\alpha +k-1} \biggl( \frac{1-t}{n+1} \biggr) ^{-s}\,\mathrm{d}t=\frac{(n+1)^{s}}{\alpha +k-s}. \end{aligned}$$

IV. If we take $h(t)=1$, then we have the result for higher order strongly P-preinvex functions.

Corollary 4.20

Suppose that $k\in \mathbb{N}$, the function $f:I\rightarrow \mathbb{R}$ is of kth order differentiable, $a,a+\eta (b,a) \in I$, $0<\eta (b,a)$, $q>1$, and $n\in \mathbb{N}$. If $|f^{(k)}|^{q}$ is a higher order strongly P-preinvex function, then

$$\begin{aligned} & \bigl\vert \mathcal{H}(k,n,\alpha ,a,b) (f) \bigr\vert \\ &\quad \leq \frac{2}{\alpha +k} \biggl[ \bigl\vert f^{(k)}(a) \bigr\vert ^{q}+ \bigl\vert f^{(k)} (b) \bigr\vert ^{q}-\frac{ \mu (k+\alpha) }{(n+1)^{2}} \bigg(\frac{n(k+\alpha+2)+1}{(k+\alpha+1)(k+\alpha+2)}\bigg) \bigl\Vert \eta (b,a) \bigr\Vert ^{\sigma } \biggr]^{ \frac{1}{q}}. \end{aligned}$$

Conclusion

In this paper, we introduce the notion of higher order strongly h-preinvex functions. As special cases we deduce some other types of higher order strongly preinvex functions. We prove an identity related to the kth order differentiable functions and Riemann–Liouville integrals. Utilizing the identity, we obtained some estimates of upper bound for the kth order differentiable functions involving Riemann–Liouville integrals via higher order strongly h-preinvex functions. We also discussed several special cases of the main results. We would like to point out here that all the results obtained in this paper continued to hold for strongly preinvex functions, indeed, which can be observed by the special case of $\sigma =2$ and $h(t)=t$. We hope that the ideas and techniques of this paper will inspire interested readers.

References

1.
Angulo, H., Giménez, J., Moros, A.M., Nikodem, K.: On strongly h-convex functions. Ann. Funct. Anal. 2(2), 85–91 (2011)
- MathSciNet
- Article
- Google Scholar
2.
Awan, M.U., Noor, M.A., Safdar, F., Noor, K.I.: Higher order strongly preinvex functions and inequalities (2019). preprint
3.
Azócar, A., Giménez, J., Nikodem, K., Sánchez, J.L.: On strongly midconvex functions. Opusc. Math. 31(1), 15–26 (2011)
- MathSciNet
- Article
- Google Scholar
4.
Barani, A., Ghazanfari, A.G., Dragomir, S.S.: Hermite–Hadamard inequality for functions whose derivatives absolute values are preinvex. J. Inequal. Appl. 2012, Article ID 247 (2012)
- MathSciNet
- Article
- Google Scholar
5.
Breckner, W.W.: Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Raumen. Publ. Inst. Math. 23, 13–20 (1978)
- MATH
- Google Scholar
6.
Craven, B.D.: Duality for generalized convex fractional programs. In: Schaible, S., Ziemba, T. (eds.) Generalized Convexity in Optimization and Economics, pp. 473–489. Academic Press, San Diego (1981)
- Google Scholar
7.
Cristescu, G., Lupsa, L.: Non-connected Convexities and Applications. Kluwer Academic, Dordrecht (2002)
- Google Scholar
8.
Cristescu, G., Noor, M.A., Awan, M.U.: Bounds of the second degree cumulative frontier gaps of functions with generalized convexity. Carpath. J. Math. 31(2), 173–180 (2015)
- MathSciNet
- MATH
- Google Scholar
9.
Dragomir, S.S.: Inequalities of Hermite–Hadamard type for h-convex functions on linear spaces. Proyecciones 34(4), 323–341 (2015)
- MathSciNet
- Article
- Google Scholar
10.
Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite–Hadamard Inequalities and Applications. Victoria University Australia (2000)
- Google Scholar
11.
Dragomir, S.S., Pečarić, J., Persson, L.E.: Some inequalities of Hadamard type. Soochow J. Math. 21(3), 335–341 (1995)
- MathSciNet
- MATH
- Google Scholar
12.
Godunova, E.K., Levin, V.I.: Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, in: Numerical Mathematics and Mathematical Physics, 138-142, 166 (Russian). Moskov. Gos. Ped. Inst., Moscow (1985)
- Google Scholar
13.
Hanson, M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)
- MathSciNet
- Article
- Google Scholar
14.
Karamardian, S.: The nonlinear complementarity problem with applications, Part 2. J. Optim. Theory Appl. 4(3), 167–181 (1969)
- MathSciNet
- Article
- Google Scholar
15.
Kilbas, A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
- Google Scholar
16.
Lin, G.H., Fukushima, M.: Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 118(1), 67–80 (2003)
- MathSciNet
- Article
- Google Scholar
17.
Merentes, N., Nikodem, K.: Remarks on strongly convex functions. Aequ. Math. 80(1–2), 193–199 (2010)
- MathSciNet
- Article
- Google Scholar
18.
Mishra, S.K., Sharma, N.: On strongly generalized convex functions of higher order. Math. Inequal. Appl. 22(1), 111–121 (2019)
- MathSciNet
- MATH
- Google Scholar
19.
Mititelu, S.: Invex sets. Stud. Cercet. Ştiinţ. - Univ. Bacău, Ser. Mat. 46(5), 529–532 (1994)
- MathSciNet
- MATH
- Google Scholar
20.
Mohan, S.R., Neogy, S.K.: On invex sets and preinvex functions. J. Math. Anal. Appl. 189, 901–908 (1995)
- MathSciNet
- Article
- Google Scholar
21.
Nikodem, K., Páles, Z.: Characterizations of inner product spaces by strongly convex functions. Banach J. Math. Anal. 5(1), 83–87 (2011)
- MathSciNet
- Article
- Google Scholar
22.
Noor, M.A.: Hermite–Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2, 126–131 (2007)
- MathSciNet
- MATH
- Google Scholar
23.
Noor, M.A., Cristescu, G., Awan, M.U.: Bounds having Riemann type quantum integrals via strongly convex functions. Studia Sci. Math. Hung. 54(2), 221–240 (2017)
- MathSciNet
- MATH
- Google Scholar
24.
Noor, M.A., Noor, K.I.: On strongly generalized preinvex functions. J. Inequal. Pure Appl. Math. 6(4), Article ID 102 (2005)
- MathSciNet
- MATH
- Google Scholar
25.
Noor, M.A., Noor, K.I., Awan, M.U., Li, J.: On Hermite–Hadamard inequalities for h-preinvex functions. Filomat 28(7), 1463–1474 (2014)
- MathSciNet
- Article
- Google Scholar
26.
Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966)
- Google Scholar
27.
Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N.: Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57, 2403–2407 (2013)
- Article
- Google Scholar
28.
Set, E., Celik, B.: On generalizations related to the left side of Fejer’s inequality via fractional integral operator. Miskolc Math. Notes 18(2), 1043–1057 (2017)
- MathSciNet
- Article
- Google Scholar
29.
Set, E., Iscan, I., Sarikaya, M.Z., Ozdemir, M.: On new inequalities of Hermite–Hadamard–Fejér type for convex functions via fractional integrals. Appl. Math. Comput. 259, 875–888 (2015)
- MathSciNet
- MATH
- Google Scholar
30.
Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326, 26–35 (2007)
- Article
- Google Scholar
31.
Weir, T., Mond, B.: Preinvex functions in multi objective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)
- MathSciNet
- Article
- Google Scholar
32.
Zhang, Y., Du, T.S., Wang, H.: Some new k-fractional integral inequalities containing multiple parameters via generalized $(s,m)$-preinvexity. Ital. J. Pure Appl. Math. 40, 510–527 (2018)
- MATH
- Google Scholar

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Acknowledgements

The authors would like to express sincere appreciation to the editors and the anonymous reviewers for their valuable comments and suggestions which helped to improve the manuscript. The second author is thankful for the support of HEC project (No. 8081/Punjab/NRPU/R&D/HEC/2017).

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The datasets used or analysed during the current study are available from the corresponding author on reasonable request.

Funding

This work was supported by the Science and Technology Project of Fujian Province Education Department of China (Grant No. JAT160485).

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Department of Mathematics, Longyan University, Longyan, China
- Shanhe Wu
Department of Mathematics, Government College University, Faisalabad, Pakistan
- Muhammad Uzair Awan
- & Sadia Talib
Department of Scientific-Methodical Sessions, Romanian Mathematical Society-Branch, Bucharest, Romania
- Marcela V. Mihai
Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
- Muhammad Aslam Noor

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Contributions

SW and MUA finished the proofs of the main results and the writing work. MVM, MAN, and ST gave lots of advice on the proofs of the main results and the writing work. All authors read and approved the final manuscript.

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Wu, S., Awan, M.U., Mihai, M.V. et al. Estimates of upper bound for a kth order differentiable functions involving Riemann–Liouville integrals via higher order strongly h-preinvex functions. J Inequal Appl 2019, 227 (2019) doi:10.1186/s13660-019-2146-z

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Received
19 February 2019
Accepted
01 July 2019
Published
28 August 2019
DOI
https://doi.org/10.1186/s13660-019-2146-z

MSC

26A51
26D15
52A01

Keywords

Estimates of upper bound
kth order differentiable functions
Strongly h-preinvex functions
Riemann–Liouville integrals

Estimates of upper bound for a kth order differentiable functions involving Riemann–Liouville integrals via higher order strongly h-preinvex functions

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Abstract

Introduction

Definition 1.1

Definition 1.2

Condition C

Definition 1.3

Definition 1.4

Theorem 1.5

Higher order strongly h-preinvex functions

Definition 2.1

Definition 2.2

Definition 2.3

Definition 2.4

Definition 2.5

Definition 2.6

Remark 2.7

Auxiliary result

Definition 3.1

Lemma 3.2

Proof

Main results

Theorem 4.1

Proof

Corollary 4.2

Corollary 4.3

Corollary 4.4

Corollary 4.5

Theorem 4.6

Proof

Corollary 4.7

Corollary 4.8

Corollary 4.9

Corollary 4.10

Theorem 4.11

Proof

Corollary 4.12

Corollary 4.13

Corollary 4.14

Corollary 4.15

Theorem 4.16

Proof

Corollary 4.17

Corollary 4.18

Corollary 4.19

Corollary 4.20

Conclusion

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Department of Mathematics, Longyan University, Longyan, China

Department of Mathematics, Government College University, Faisalabad, Pakistan

Department of Scientific-Methodical Sessions, Romanian Mathematical Society-Branch, Bucharest, Romania

Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan

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