Skip to main content

Fractional inclusions of the Hermite–Hadamard type for m-polynomial convex interval-valued functions

Abstract

The notion of m-polynomial convex interval-valued function \(\Psi =[\psi ^{-}, \psi ^{+}]\) is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions \(\psi ^{-}\) and \(\psi ^{+}\). For this class of functions, we establish loads of new set inclusions of the Hermite–Hadamard type involving the ρ-Riemann–Liouville fractional integral operators. In particular, we prove, among other things, that if a set-valued function Ψ defined on a convex set S is m-polynomial convex, \(\rho,\epsilon >0\) and \(\zeta,\eta \in {\mathbf{S}}\), then

$$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)& \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ), \end{aligned}$$

where Ψ is Lebesgue integrable on \([\zeta,\eta ]\), \(S_{p}(\epsilon;\rho )=2-\frac{\epsilon }{\epsilon +\rho p}- \frac{\epsilon }{\rho }\mathcal{B} (\frac{\epsilon }{\rho }, p+1 )\) and \(\mathcal{B}\) is the beta function. We extend, generalize, and complement existing results in the literature. By taking \(m\geq 2\), we derive loads of new and interesting inclusions. We hope that the idea and results obtained herein will be a catalyst towards further investigation.

Introduction

A set \({\mathbf{S}}\subset \mathbb{R}\) is called a convex set if \(\xi w+(1-\xi )z\in {\mathbf{S}}\) for all \(w,z\in {\mathbf{S}}\) and \(\xi \in [0,1]\). We call a function \(\psi:{\mathbf{S}}\to \mathbb{R}\) convex if

$$\begin{aligned} \psi \bigl(\xi w+(1-\xi )z \bigr)\leq \xi \psi (z)+(1-\xi )\psi (w) \end{aligned}$$

for all \(w,z\in {\mathbf{S}}\) and \(\xi \in [0,1]\). It is generally known that if \(\psi:[\zeta,\eta ]\to \mathbb{R}\) is convex, then

$$\begin{aligned} \psi \biggl(\frac{\zeta +\eta }{2} \biggr)\leq \frac{1}{\eta -\zeta } \int _{\zeta }^{\eta }\psi (r) \,dr\leq \frac{\psi (\zeta )+\psi (\eta )}{2}. \end{aligned}$$
(1)

Inequality (1) is today known as the Hermite–Hadamard inequality. It was named after two French mathematicians, Charles Hermite and Jacques Hadamard. The former [17] first established the result in 1883, and a decade later it was rediscovered by the latter [16].

There are loads of articles in the literature on generalizations and extensions of (1) for different kinds of convexities. Examples of such can be found in [15, 10, 11, 14, 15, 1826, 33, 34, 38] and the references cited therein. Recently, Toplu et al. [39] proposed and defined an m-polynomial convex function as follows: a real-valued function \(\psi:{\mathbf{S}}\to {\mathbb{R}}^{+}:=(0,\infty )\) is m-polynomial convex (concave) if

$$\begin{aligned} \psi \bigl(\xi w+(1-\xi )z \bigr)\leq (\geq ) \frac{1}{m}\sum _{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr]\psi (w)+\frac{1}{m}\sum_{p=1}^{m} \bigl[1-{\xi }^{p} \bigr]\psi (z) \end{aligned}$$

for all \(w,z\in {\mathbf{S}}\) and \(\xi \in [0,1]\). In this paper, we shall denote the sets of all m-polynomial convex and m-polynomial concave functions from S into \({\mathbb{R}}^{+}\) by \({\mathbf{XP}}_{m} ({\mathbf{S}},{\mathbb{R}}^{+} )\) and \({\mathbf{VP}}_{m} ({\mathbf{S}},{\mathbb{R}}^{+} )\), respectively. In the same paper, the authors established the following Hermite–Hadamard type inequality for this class of functions.

Theorem 1

([39])

Let \(\psi:[\zeta,\eta ]\to {\mathbb{R}}^{+}\)be an m-polynomial convex function. If \(\zeta <\eta \)and ψ is Lebesgue integrable on \([\zeta,\eta ]\), then

$$\begin{aligned} \frac{2^{-1} m}{m+2^{-m}-1}\psi \biggl(\frac{\zeta +\eta }{2} \biggr) \leq \frac{1}{\eta -\zeta } \int _{\zeta }^{\eta }\psi (r) \,dr\leq \frac{\psi (\zeta )+\psi (\eta )}{m} \sum_{p=1}^{m}\frac{p}{p+1}. \end{aligned}$$
(2)

Now, recall that the left- and right-sided ρ-Riemann–Liouville fractional integral operators \({}_{\rho }{\mathcal{J}}_{\zeta ^{+}}^{\epsilon }\) and \({}_{\rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\) of order \(\epsilon >0\), for a real-valued continuous function \(\psi (w)\), are defined as follows:

$$\begin{aligned} _{\rho }{\mathcal{J}}_{\zeta ^{+}}^{\epsilon }\psi (w) = \frac{1}{\rho \Gamma _{\rho }(\epsilon )} \int _{\zeta }^{w}(w-\xi )^{ \frac{\epsilon }{\rho }-1}\psi (\xi ) \,d \xi,\quad w>\zeta, \end{aligned}$$
(3)

and

$$\begin{aligned} _{k}{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\psi (w) = \frac{1}{\rho \Gamma _{\rho }(\epsilon )} \int _{w}^{\eta } (\xi -w )^{\frac{\epsilon }{\rho }-1}\psi (\xi ) \,d\xi,\quad w< \eta, \end{aligned}$$
(4)

where \(\rho >0\), and \(\Gamma _{\rho }\) is the ρ-gamma function given by

$$\begin{aligned} \Gamma _{\rho }(w):= \int _{0}^{\infty }\xi ^{w-1}e^{- \frac{\xi ^{\rho }}{\rho }} \,d\xi,\qquad \operatorname{Re}(w)>0, \end{aligned}$$

with the properties \(\Gamma _{\rho }(w+\rho )=w\Gamma _{\rho }(w)\) and \(\Gamma _{\rho }(\rho )=1\). If \(\rho =1\), we simply write

$$\begin{aligned} {}_{1}{\mathcal{J}}_{\zeta ^{+}}^{\epsilon }\psi (w)={ \mathcal{J}}_{ \zeta ^{+}}^{\epsilon }\psi (w) \quad\text{and}\quad {}_{1}{ \mathcal{J}}_{ \eta ^{-}}^{\epsilon }\psi (w)={\mathcal{J}}_{\eta ^{-}}^{\epsilon } \psi (w). \end{aligned}$$

The beta function \(\mathcal{B}\) is defined by

$$\begin{aligned} \mathcal{B}(u,v)= \int _{0}^{1}\xi ^{u-1}(1-\xi )^{v-1} \,d\xi\quad \text{for } \operatorname{Re}(u)>0, \operatorname{Re}(v)>0. \end{aligned}$$
(5)

Using these fractional integral operators, Sarikaya et al. [37] established the following fractional version of (1).

Theorem 2

([37])

Let \(\psi:[\zeta,\eta ]\to {\mathbb{R}}^{+}\)be a convex function. If \(0\leq \zeta <\eta \)and ψ is Lebesgue integrable on \([\zeta,\eta ]\), then the following double inequalities for the Riemann–Liouville fractional integrals hold:

$$\begin{aligned} \psi \biggl(\frac{\zeta +\eta }{2} \biggr)\leq \frac{\Gamma (\epsilon +1)}{2(\eta -\zeta )^{\epsilon }} \bigl[{ \mathcal{J}}_{\zeta ^{+}}^{\epsilon }\psi (\eta )+{ \mathcal{J}}_{ \eta ^{-}}^{\epsilon }\psi (\zeta ) \bigr]\leq \frac{\psi (\zeta )+\psi (\eta )}{2}, \end{aligned}$$
(6)

where \(\epsilon >0\).

The theory of interval analysis [29] was initiated by the late American mathematician Ramon E. Moore in 1966. Since its advent, this field has received ample amount of attention from different researchers in the mathematical community. Experts have found applications of interval analysis in global optimization and constraint solution algorithms. It has since grown steadily in popularity over the past decades. Interval analysis has been found to be valuable to engineers and scientists interested in scientific computation, especially in reliability, effects of round-off error, and automatic verification of results, see [8, 9, 12, 13]. With the birth of interval analysis, mathematicians, those who work in the field of mathematical inequalities, want to know if the inequalities in the above-mentioned results can be replaced with inclusions. In some cases, the answer to the question is in the affirmative. In this light, Sadowska (see also [28]) established the following result for a given interval-valued function.

Theorem 3

([36])

Let Ψ be a nonnegative continuous convex set-valued function on \([\zeta,\eta ]\). Then

$$\begin{aligned} \Psi \biggl(\frac{\zeta +\eta }{2} \biggr)\supset \frac{1}{\eta -\zeta } \int _{\zeta }^{\eta }\Psi (r) \,dr\supset \frac{\Psi (\zeta )+\Psi (\eta )}{2}. \end{aligned}$$
(7)

Results related to (7), for different families of set-valued convex functions, have been established. For example, see the papers [6, 8, 9, 12, 13, 27, 32, 35, 40, 41]. Recently, Budak et al. [7] established the following interval counterpart of (6).

Theorem 4

([7])

Let Ψ be a convex interval-valued function defined on \([\zeta,\eta ]\)such that \(\Psi =[\psi ^{-}, \psi ^{+}]\). If \(0 \leq \zeta <\eta \)and \(\epsilon >0\), then

$$\begin{aligned} \Psi \biggl(\frac{\zeta +\eta }{2} \biggr)\supseteq \frac{\Gamma (\epsilon +1)}{2(\eta -\zeta )^{\epsilon }} \bigl[{ \mathcal{J}}_{\zeta ^{+}}^{\epsilon }\Psi (\eta )+{ \mathcal{J}}_{ \eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr]\supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{2}. \end{aligned}$$
(8)

This work is inspired by the above-mentioned articles. It is our purpose in this article to propose a new class of interval-valued functions called the m-polynomial convex functions and then obtain the interval-valued counterpart of (2). This result involves the ρ-Riemann–Liouville fractional integral operators and generalizes Theorem 4. In addition, we establish four more results in this direction. Our results complement and extend known results in [7] and others in the literature. The paper is arranged as follows: in Sect. 2, we present a quick overview of the theory of interval analysis. Section 3 contains our main results with detailed justifications. Interesting corollaries are also pointed out. A brief introduction follows thereafter.

Preliminaries

Interval analysis is roughly described as an analysis of interval-valued functions. It is an annex of numerical analysis where instead of real numbers intervals are used as its operating element. In this section, we collate some basic terms and essentials of the theory of interval analysis from the books [2931]. In the sequel, let \({\mathbb{K}}_{c}\) represent the class of all bounded closed nonempty intervals in \(\mathbb{R}\), i.e.,

$$\begin{aligned} {\mathbb{K}}_{c}:= \bigl\{ \bigl[\zeta ^{-},\zeta ^{+} \bigr]| \zeta ^{-},\zeta ^{+}\in \mathbb{R} \text{ and } \zeta ^{-} \leq \zeta ^{+} \bigr\} . \end{aligned}$$

The numbers \(\zeta ^{-}\) and \(\zeta ^{+}\) are called the left and right endpoints of \([\zeta ^{-},\zeta ^{+} ]\), respectively. The interval \([\zeta ^{-},\zeta ^{+} ]\) is called degenerated if \(\zeta ^{-}=\zeta ^{+}\); positive if \(\zeta ^{-}>0\); and negative if \(\zeta ^{+}<0\). We denote the sets of all negative intervals and positive intervals in \(\mathbb{R}\) by \({\mathbb{K}}_{c}^{-}\) and \({\mathbb{K}}_{c}^{+}\), respectively. That is,

$$\begin{aligned} {\mathbb{K}}_{c}^{-}:= \bigl\{ \bigl[\zeta ^{-}, \zeta ^{+} \bigr] \in {\mathbb{K}}_{c} | \zeta ^{+}< 0 \bigr\} \end{aligned}$$

and

$$\begin{aligned} {\mathbb{K}}_{c}^{+}:= \bigl\{ \bigl[\zeta ^{-}, \zeta ^{+} \bigr] \in {\mathbb{K}}_{c} | \zeta ^{-}>0 \bigr\} . \end{aligned}$$

Let \(A= [\zeta ^{-},\zeta ^{+} ]\), \(B= [\eta ^{-},\eta ^{+} ]\in {\mathbb{K}}_{c}\), and \(\gamma \in \mathbb{R}\). We say \(A\subseteq B\) (or \(B\supseteq A\)) if and only if \(\eta ^{-}\leq \zeta ^{-}\) and \(\zeta ^{+}\leq \eta ^{+}\). The following arithmetic operations are defined thus:

$$\begin{aligned} &\gamma A= \textstyle\begin{cases} [\gamma \zeta ^{-},\gamma \zeta ^{+} ]& \text{if } \gamma >0, \\ \{0\}& \text{if } \gamma =0, \\ [\gamma \zeta ^{+},\gamma \zeta ^{-} ]& \text{if } \gamma < 0; \end{cases}\displaystyle \\ &A+B= \bigl[\zeta ^{-},\zeta ^{+} \bigr]+ \bigl[\eta ^{-},\eta ^{+} \bigr]:= \bigl[\zeta ^{-}+\eta ^{-},\zeta ^{+}+\eta ^{+} \bigr]; \\ &A-B= \bigl[\zeta ^{-},\zeta ^{+} \bigr]- \bigl[\eta ^{-},\eta ^{+} \bigr]:= \bigl[\zeta ^{-}-\eta ^{+},\zeta ^{+}-\eta ^{-} \bigr]; \\ &A\cdot B:= \bigl[\min \bigl\{ \zeta ^{-}\eta ^{-},\zeta ^{-}\eta ^{+}, \zeta ^{+}\eta ^{-}, \zeta ^{+}\eta ^{+} \bigr\} ,\max \bigl\{ \zeta ^{-} \eta ^{-},\zeta ^{-}\eta ^{+}, \zeta ^{+}\eta ^{-}, \zeta ^{+}\eta ^{+} \bigr\} \bigr]; \\ &\frac{A}{B}:= \biggl[\min \biggl\{ \frac{\zeta ^{-}}{\eta ^{-}}, \frac{\zeta ^{-}}{\eta ^{+}}, \frac{\zeta ^{+}}{\eta ^{-}}, \frac{\zeta ^{+}}{\eta ^{+}} \biggr\} ,\max \biggl\{ \frac{\zeta ^{-}}{\eta ^{-}},\frac{\zeta ^{-}}{\eta ^{+}}, \frac{\zeta ^{+}}{\eta ^{-}}, \frac{\zeta ^{+}}{\eta ^{+}} \biggr\} \biggr];\quad 0\notin B. \end{aligned}$$

Interval addition is commutative, associative and \({\mathbf{0}}=[0,0]\) is the identity element. Additive inverses do not exist, but the cancelation law holds. Also, interval multiplication is commutative, associative and \({\mathbf{1}}=[1,1]\) is the identity element. Multiplicative inverses do not exist and the cancelation law does not hold either. The distributive rule is not valid in general. It is important to also note that the interval arithmetic is said to be inclusion isotonic (see [31, p. 34]). By this, we mean that if A, B, C, and D are intervals such that

$$\begin{aligned} A\subseteq B \quad\text{and}\quad C\subseteq D, \end{aligned}$$

then

$$\begin{aligned} A\boxtimes C\subseteq B\boxtimes D, \end{aligned}$$

where stands for interval addition, subtraction, multiplication, or division. It follows therefore that if \(\zeta \leq \eta \) and \(C\subseteq D\), then with \(A=[\zeta,\zeta ]\) and \(B=[\eta,\eta ]\), we have that \(\zeta C\subseteq \eta D\).

The Pompeiu–Hausdorff distance \(d_{H}:{\mathbb{K}}_{c}\times {\mathbb{K}}_{c}\to {\mathbb{R}}_{+} \cup \{0\}\) is defined by

$$\begin{aligned} d_{H}:=\max \Bigl\{ \max_{\zeta \in A} \,d(\zeta, B), \max _{ \eta \in B} \,d(\eta, A) \Bigr\} \quad\text{with } \,d(\eta, A)= \min _{\zeta \in A} \vert \eta -\zeta \vert . \end{aligned}$$

It is generally known that \(({\mathbb{K}}_{c}, d_{H})\) is a complete metric space. The concept of a convergent sequence of intervals \((A_{n})_{n\in \mathbb{N}}\), \(A_{n}\in {\mathbb{K}}_{c}\) is considered in the complete metric space \({\mathbb{K}}_{c}\), endowed with the \(d_{H}\) distance: We say that \(\lim_{n\to \infty }A_{n}=A\) if and only if for any real number \(\epsilon >0\) there exists \(N_{\epsilon }\in \mathbb{N}\) such that

$$\begin{aligned} d_{H}(A_{n},A)< \epsilon \quad\text{for all } n>N_{\epsilon }. \end{aligned}$$

Next, we turn our attention to interval-valued functions.

Definition 5

An interval-valued function is defined to be any \(\Psi:[\zeta,\eta ]\to {\mathbb{K}}_{c}\) with \(\Psi (w)= [\psi ^{-}(w),\psi ^{+}(w) ]\in {\mathbb{K}}_{c}\) and \(\psi ^{-}(w)\leq \psi ^{+}(w)\) for all \(w\in [\zeta,\eta ]\). We say that Ψ is Lebesgue integrable on \([\zeta,\eta ]\) if the real-valued functions \(\psi ^{-}\) and \(\psi ^{+}\) are Lebesgue integrable on \([\zeta,\eta ]\), and then we write

$$\begin{aligned} \int _{\zeta }^{\eta }\Psi (r) \,dr= \biggl[ \int _{\zeta }^{\eta }\psi ^{-}(r) \,dr, \int _{\zeta }^{\eta }\psi ^{+}(r) \,dr \biggr]. \end{aligned}$$

For an interval function \(\Psi (w)= [\psi ^{-}(w),\psi ^{+}(w) ]\), we define the ρ-Riemann–Liouville integral operators as follows:

$$\begin{aligned} _{\rho }{\mathcal{J}}_{\zeta ^{+}}^{\epsilon }\Psi (w)= \bigl[{_{\rho }{ \mathcal{J}}}_{\zeta ^{+}}^{\epsilon }\psi ^{-}(w), _{\rho }{ \mathcal{J}}_{\zeta ^{+}}^{\epsilon }\psi ^{+}(w) \bigr] \end{aligned}$$

and

$$\begin{aligned} _{\rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (w)= \bigl[ {_{\rho }{ \mathcal{J}}}_{\eta ^{-}}^{\epsilon }\psi ^{-}(w), _{\rho }{ \mathcal{J}}_{\eta ^{-}}^{\epsilon }\psi ^{+}(w) \bigr]. \end{aligned}$$

Main results

We first introduce the notion of m-polynomial convex interval-valued function.

Definition 6

Let S be a convex set, \(\Psi:{\mathbf{S}}\to {\mathbb{K}}_{c}^{+}\) be an interval-valued function, and \(m\in \mathbb{N}\). We say that Ψ is m-polynomial convex (concave) if and only if

$$\begin{aligned} \frac{1}{m}\sum_{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr]\Psi (w)+ \frac{1}{m}\sum _{p=1}^{m} \bigl[1-{\xi }^{p} \bigr]\Psi (z) \subseteq (\supseteq ) \Psi \bigl(\xi w+(1-\xi )z \bigr) \end{aligned}$$
(9)

for all \(w,z\in {\mathbf{S}}\) and \(\xi \in [0,1]\). In what follows, we shall denote the sets of all m-polynomial convex and m-polynomial concave interval-valued functions from S into \({\mathbb{K}}_{c}^{+}\) by \({\mathbf{XP}}_{m} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\) and \({\mathbf{VP}}_{m} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\), respectively.

Remark 7

If we take a particular value of m, then we get a corresponding set inclusion. Take, for instance:

  1. 1.

    If \(m=1\), then we get the definition of a convex interval-valued function

    $$\begin{aligned} \Psi \bigl(\xi w+(1-\xi )z \bigr)\supseteq \xi \Psi (w)+(1-\xi )\Psi (z) \end{aligned}$$

    for all \(w,z\in {\mathbf{S}}\) and \(\xi \in [0,1]\);

  2. 2.

    For \(m=2\), we get the following inclusion for a 2-polynomial convex interval-valued function:

    $$\begin{aligned} \Psi \bigl(\xi w+(1-\xi )z \bigr)\supseteq \frac{3\xi -\xi ^{2}}{2}\Psi (w)+ \frac{2-\xi -\xi ^{2}}{2}\Psi (z) \end{aligned}$$

    for all \(w,z\in {\mathbf{S}}\) and \(\xi \in [0,1]\);

  3. 3.

    For \(m=3\), we deduce the succeeding relation for a 3-polynomial convex interval-valued function:

    $$\begin{aligned} \Psi \bigl(\xi w+(1-\xi )z \bigr)\supseteq \frac{6\xi -4\xi ^{2}+\xi ^{3}}{3}\Psi (w)+ \frac{3-\xi -\xi ^{2}-\xi ^{3}}{3}\Psi (z) \end{aligned}$$

    for all \(w,z\in {\mathbf{S}}\) and \(\xi \in [0,1]\).

We now present a theorem that gives a link between a given interval-valued function Ψ and its component real-valued functions \(\psi ^{-}\) and \(\psi ^{+}\).

Theorem 8

Let \(\Psi:{\mathbf{S}}\to {\mathbb{K}}_{c}^{+}\)be an interval-valued function such that \(\Psi (w)= [\psi ^{-}(w), \psi ^{+}(w) ]\in {\mathbb{K}}_{c}\), and \(\psi ^{-}(w)\leq \psi ^{+}(w)\)for all \(w\in [\zeta,\eta ]\). Then \(\Psi \in {\mathbf{XP}}_{m} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\)if and only if \(\psi ^{-}\in {\mathbf{XP}}_{m} ({\mathbf{S}},{\mathbb{R}}^{+} )\)and \(\psi ^{+}\in {\mathbf{VP}}_{m} ({\mathbf{S}},{\mathbb{R}}^{+} )\).

Proof

Let \(w,z\in {\mathbf{S}}\) and \(\xi \in [0,1]\). Then

$$\begin{aligned} \Psi \in {\mathbf{XP}}_{m} \bigl({\mathbf{S}},{\mathbb{K}}_{c}^{+} \bigr) \end{aligned}$$

if and only if

$$\begin{aligned} \frac{1}{m}\sum_{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr]\Psi (w)+ \frac{1}{m}\sum _{p=1}^{m} \bigl[1-{\xi }^{p} \bigr]\Psi (z) \subseteq \Psi \bigl(\xi w+(1-\xi )z \bigr) \end{aligned}$$

if and only if

$$\begin{aligned} & \Biggl[\frac{1}{m}\sum_{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr]\psi ^{-}(w)+ \frac{1}{m} \sum_{p=1}^{m} \bigl[1-{\xi }^{p} \bigr]\psi ^{-}(z), \\ &\qquad \frac{1}{m}\sum_{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr] \psi ^{+}(w)+\frac{1}{m} \sum_{p=1}^{m} \bigl[1-{\xi }^{p} \bigr] \psi ^{+}(z) \Biggr] \\ &\quad\subseteq \bigl[\psi ^{-} \bigl(\xi w+(1-\xi )z \bigr), \psi ^{+} \bigl( \xi w+(1-\xi )z \bigr) \bigr] \end{aligned}$$

if and only if

$$\begin{aligned} \frac{1}{m}\sum_{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr]\psi ^{-}(w)+ \frac{1}{m} \sum_{p=1}^{m} \bigl[1-{\xi }^{p} \bigr]\psi ^{-}(z)\geq \psi ^{-} \bigl(\xi w+(1-\xi )z \bigr), \end{aligned}$$

and

$$\begin{aligned} \frac{1}{m}\sum_{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr]\psi ^{+}(w)+ \frac{1}{m} \sum_{p=1}^{m} \bigl[1-{\xi }^{p} \bigr]\psi ^{+}(z)\leq \psi ^{+} \bigl(\xi w+(1-\xi )z \bigr) \end{aligned}$$

if and only if

$$\begin{aligned} \psi ^{-}\in {\mathbf{XP}}_{m} \bigl({\mathbf{S}},{ \mathbb{R}}^{+} \bigr) \quad\text{and}\quad \psi ^{+}\in { \mathbf{VP}}_{m} \bigl({\mathbf{S}},{\mathbb{R}}^{+} \bigr). \end{aligned}$$

That completes the proof in both directions. □

In a similar manner, one can prove the following result.

Theorem 9

Let \(\Psi:{\mathbf{S}}\to {\mathbb{K}}_{c}^{+}\)be an interval-valued function such that \(\Psi (w)= [\psi ^{-}(w), \psi ^{+}(w) ]\in {\mathbb{K}}_{c}\)and \(\psi ^{-}(w)\leq \psi ^{+}(w)\)for all \(w\in [\zeta,\eta ]\). Then \(\Psi \in {\mathbf{VP}}_{m} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\)if and only if \(\psi ^{-}\in {\mathbf{VP}}_{m} ({\mathbf{S}},{\mathbb{R}}^{+} )\)and \(\psi ^{+}\in {\mathbf{XP}}_{m} ({\mathbf{S}},{\mathbb{R}}^{+} )\).

For the remaining part of this article, we shall assume that \(\Psi:{\mathbf{S}}\to {\mathbb{K}}_{c}^{+}\) is always of the form \(\Psi (w)= [\psi ^{-}(w),\psi ^{+}(w) ]\in {\mathbb{K}}_{c}\) and \(\psi ^{-}(w)\leq \psi ^{+}(w)\) for all \(w\in [\zeta,\eta ]\). We are now ready to formulate and prove some Hermite–Hadamard type results for m-polynomial convex (concave) interval-valued functions.

Theorem 10

Let \(\Psi:{\mathbf{S}}\to {\mathbb{K}}_{c}^{+}\)be an interval-valued function with \(\zeta <\eta \)and \(\zeta,\eta \in {\mathbf{S}}\), and Lebesgue integrable on \([\zeta,\eta ]\). If \(\Psi \in {\mathbf{XP}}_{m} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\)and \(\rho, \epsilon >0\), then

$$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)& \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ), \end{aligned}$$
(10)

where

$$\begin{aligned} S_{p}(\epsilon;\rho )=2-\frac{\epsilon }{\epsilon +\rho p}- \frac{\epsilon }{\rho } \mathcal{B} \biggl(\frac{\epsilon }{\rho }, p+1 \biggr) \end{aligned}$$

and \(\mathcal{B}\)is the beta function defined by (5). The inclusions are reversed if \(\Psi \in {\mathbf{VP}}_{m} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\).

Proof

Assuming \(\Psi \in {\mathbf{XP}}_{m} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\), we get from (9) the following relation:

$$\begin{aligned} \Psi \biggl(\frac{w+z}{2} \biggr)\supseteq \frac{1}{m}\sum _{p=1}^{m} \biggl[1-\frac{1}{2^{p}} \biggr]\Psi (w)+\frac{1}{m}\sum_{p=1}^{m} \biggl[1-\frac{1}{2^{p}} \biggr]\Psi (z). \end{aligned}$$

This implies that, for all \(w,z\in {\mathbf{S}}\),

$$\begin{aligned} \frac{1}{m}\sum_{p=1}^{m} \biggl[1-\frac{1}{2^{p}} \biggr] \bigl(\Psi (w)+ \Psi (z) \bigr)\subseteq \Psi \biggl(\frac{w+z}{2} \biggr). \end{aligned}$$
(11)

Now, let \(w=\xi \zeta +(1-\xi )\eta \) and \(z=\xi \eta +(1-\xi )\zeta \) with \(\xi \in [0,1]\). Then (11) becomes

$$\begin{aligned} \frac{1}{m}\sum_{p=1}^{m} \biggl(1-\frac{1}{2^{p}} \biggr) \bigl\{ \Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr)+\Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) \bigr\} \subseteq \Psi \biggl( \frac{\zeta +\eta }{2} \biggr). \end{aligned}$$
(12)

Multiplying both sides of (12) by \(\xi ^{\frac{\epsilon }{\rho }-1}\) and then integrating with respect to ξ over \([0,1]\), we get

$$\begin{aligned} & \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \Psi \biggl( \frac{\zeta +\eta }{2} \biggr) \,d\xi \\ &\quad\supseteq \frac{1}{m}\sum_{p=1}^{m} \biggl(1-\frac{1}{2^{p}} \biggr) \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \bigl\{ \Psi \bigl( \xi \zeta +(1-\xi )\eta \bigr)+\Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) \bigr\} \,d\xi \\ &\quad=\frac{1}{m}\sum_{p=1}^{m} \biggl(1-\frac{1}{2^{p}} \biggr) \biggl[ \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \bigl\{ \psi ^{-} \bigl(\xi \zeta +(1-\xi )\eta \bigr)+\psi ^{-} \bigl(\xi \eta +(1-\xi )\zeta \bigr) \bigr\} \,d\xi, \\ &\qquad \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \bigl\{ \psi ^{+} \bigl(\xi \zeta +(1-\xi )\eta \bigr)+\psi ^{+} \bigl(\xi \eta +(1-\xi )\zeta \bigr) \bigr\} \,d\xi \biggr]. \end{aligned}$$
(13)

Now,

$$\begin{aligned} & \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \bigl\{ \psi ^{-} \bigl(\xi \zeta +(1-\xi )\eta \bigr)+\psi ^{-} \bigl(\xi \eta +(1-\xi )\zeta \bigr) \bigr\} \,d\xi \\ &\quad=\frac{1}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \biggl[ \int _{ \zeta }^{\eta }(\eta -r)^{\frac{\epsilon }{\rho }-1}\psi ^{-}(r) \,dr + \int _{\zeta }^{\eta }(r-\zeta )^{\frac{\epsilon }{\rho }-1}\psi ^{-}(r) \,dr \biggr] \\ &\quad= \frac{\rho \Gamma _{\rho }(\epsilon )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \biggl[\frac{1}{\rho \Gamma _{\rho }(\epsilon )} \int _{\zeta }^{\eta }( \eta -r)^{\frac{\epsilon }{\rho }-1}\psi ^{-}(r) \,dr + \frac{1}{\rho \Gamma _{\rho }(\epsilon )} \int _{\zeta }^{\eta }(r-\zeta )^{ \frac{\epsilon }{\rho }-1}\psi ^{-}(r) \,dr \biggr] \\ &\quad= \frac{\rho \Gamma _{\rho }(\epsilon )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \psi ^{-}(\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon } \psi ^{-}(\zeta ) \bigr]. \end{aligned}$$
(14)

Similarly, one obtains that

$$\begin{aligned} & \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \bigl\{ \psi ^{+} \bigl(\xi \zeta +(1-\xi )\eta \bigr)+\psi ^{+} \bigl(\xi \eta +(1-\xi )\zeta \bigr) \bigr\} \,d\xi \\ &\quad= \frac{\rho \Gamma _{\rho }(\epsilon )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \psi ^{+}(\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon } \psi ^{+}(\zeta ) \bigr]. \end{aligned}$$
(15)

On the other hand,

$$\begin{aligned} \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \Psi \biggl( \frac{\zeta +\eta }{2} \biggr) \,d\xi &= \biggl[ \int _{0}^{1}\xi ^{ \frac{\epsilon }{\rho }-1}\psi ^{-} \biggl(\frac{\zeta +\eta }{2} \biggr) \,d\xi, \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1}\psi ^{+} \biggl(\frac{\zeta +\eta }{2} \biggr) \,d\xi \biggr] \\ &= \biggl[\frac{\rho }{\epsilon }\psi ^{-} \biggl(\frac{\zeta +\eta }{2} \biggr), \frac{\rho }{\epsilon }\psi ^{+} \biggl( \frac{\zeta +\eta }{2} \biggr) \biggr] \\ &=\frac{\rho }{\epsilon }\Psi \biggl(\frac{\zeta +\eta }{2} \biggr). \end{aligned}$$
(16)

Using (14), (15), and (16) in (13), one gets

$$\begin{aligned} &\frac{\rho }{\epsilon }\Psi \biggl(\frac{\zeta +\eta }{2} \biggr) \\ &\quad\supseteq \frac{m+2^{-m}-1}{m} \frac{\rho \Gamma _{\rho }(\epsilon )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{ \mathcal{J}}}_{\zeta ^{+}}^{\epsilon }\psi ^{-}(\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\psi ^{-}( \zeta ), {_{ \rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon }\psi ^{+}(\eta )+_{\rho }{ \mathcal{J}}_{\eta ^{-}}^{\epsilon } \psi ^{+}(\zeta ) \bigr] \\ &\quad=\frac{m+2^{-m}-1}{m} \frac{\rho \Gamma _{\rho }(\epsilon )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl( \bigl[{_{\rho }{ \mathcal{J}}}_{\zeta ^{+}}^{\epsilon }\psi ^{-}( \eta ), {_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon }\psi ^{+}( \eta ) \bigr]+ \bigl[{_{\rho }{\mathcal{J}}}_{\eta ^{-}}^{\epsilon } \psi ^{-}( \zeta ), {_{\rho }{\mathcal{J}}}_{\eta ^{-}}^{\epsilon } \psi ^{+}( \zeta ) \bigr] \bigr) \\ &\quad=\frac{m+2^{-m}-1}{m} \frac{\rho \Gamma _{\rho }(\epsilon )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{ \mathcal{J}}}_{\zeta ^{+}}^{\epsilon }\Psi (\eta )+_{ \rho }{ \mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr]. \end{aligned}$$

This further implies that

$$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr) \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr]. \end{aligned}$$
(17)

Next, we get from (9) the following inclusions:

$$\begin{aligned} &\Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr) \\ &\quad\supseteq \frac{1}{m}\sum_{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr] \Psi (\zeta )+\frac{1}{m}\sum _{p=1}^{m} \bigl[1-{\xi }^{p} \bigr] \Psi (\eta ) \end{aligned}$$
(18)

and

$$\begin{aligned} &\Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) \\ &\quad\supseteq \frac{1}{m}\sum_{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr] \Psi (\eta )+\frac{1}{m}\sum _{p=1}^{m} \bigl[1-{\xi }^{p} \bigr] \Psi (\zeta ). \end{aligned}$$
(19)

Adding (18) and (19) gives

$$\begin{aligned} &\Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr)+\Psi \bigl(\xi \eta +(1-\xi ) \zeta \bigr) \\ &\quad\supseteq \frac{1}{m} \Biggl\{ \sum_{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr]+\sum_{p=1}^{m} \bigl[1-{\xi }^{p} \bigr] \Biggr\} \bigl(\Psi ( \zeta )+\Psi (\eta ) \bigr). \end{aligned}$$
(20)

Multiplying (20) by \(\xi ^{\frac{\epsilon }{\rho }-1}\) and integrating the resulting inclusion with respect to ξ over \([0,1]\), we obtain

$$\begin{aligned} &\frac{\rho \Gamma _{\rho }(\epsilon )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{ \mathcal{J}}}_{\zeta ^{+}}^{\epsilon }\Psi (\eta )+_{ \rho }{ \mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ &\quad= \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \bigl[\Psi \bigl( \xi \zeta +(1-\xi )\eta \bigr)+\Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) \bigr] \,d\xi \\ &\quad\supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m} \int _{0}^{1}\xi ^{ \frac{\epsilon }{\rho }-1} \Biggl\{ \sum _{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr]+\sum_{p=1}^{m} \bigl[1-{\xi }^{p} \bigr] \Biggr\} \,d\xi \\ &\quad=\frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m} \biggl[ \frac{2\rho }{\epsilon }-\frac{\rho }{\epsilon +\rho p}-\mathcal{B} \biggl(\frac{\epsilon }{\rho }, p+1 \biggr) \biggr]. \end{aligned}$$
(21)

From (21), we obtain

$$\begin{aligned} &\frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{ \mathcal{J}}}_{\zeta ^{+}}^{\epsilon }\Psi (\eta )+_{ \rho }{ \mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ &\quad\supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m} \biggl[2- \frac{\epsilon }{\epsilon +\rho p}-\frac{\epsilon }{\rho }\mathcal{B} \biggl(\frac{\epsilon }{\rho }, p+1 \biggr) \biggr]. \end{aligned}$$
(22)

We get the intended result by combining (17) and (22). □

Remark 11

Using Theorem 10, we obtain the following particular cases:

  1. 1.

    For \(m=1\), we deduce the result for convex interval-valued functions

    $$\begin{aligned} \Psi \biggl(\frac{\zeta +\eta }{2} \biggr)\supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{2(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{2}. \end{aligned}$$
    (23)

    If, in addition, we set \(\rho =1\) in (23), then we recapture (8).

  2. 2.

    If \(m=2\), then we obtain the result for 2-polynomial convex interval-valued functions

    $$\begin{aligned} \frac{1}{5}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)&\supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{8(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{8} \biggl[1+ \frac{\epsilon }{\epsilon +\rho }-\frac{\epsilon }{\epsilon +2\rho } \biggr]. \end{aligned}$$

Theorem 12

Let \(\Psi, G:{\mathbf{S}}\to {\mathbb{K}}_{c}^{+}\)be two interval-valued functions with \(\zeta <\eta \)and \(\zeta,\eta \in {\mathbf{S}}\), and suppose that ΨG is Lebesgue integrable on \([\zeta,\eta ]\). If \(\rho, \epsilon >0\), \(\Psi \in {\mathbf{XP}}_{m_{1}} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\), and \(G\in {\mathbf{XP}}_{m_{2}} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\), then

$$\begin{aligned} &\frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )G( \eta )+_{\rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon } \Psi (\zeta )G( \zeta ) \bigr] \\ &\quad\supseteq \frac{\epsilon }{\rho } \biggl\{ \mathcal{P}(\zeta,\eta ) \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \bigl[\Delta _{1}(\xi )+ \Delta _{4}(\xi ) \bigr] \,d\xi\\ &\qquad{} +\mathcal{Q}( \zeta,\eta ) \int _{0}^{1} \xi ^{\frac{\epsilon }{\rho }-1} \bigl[\Delta _{2}(\xi )+\Delta _{3}( \xi ) \bigr] \,d\xi \biggr\} , \end{aligned}$$

where \(\mathcal{P}(\zeta,\eta )=\Psi (\zeta )G(\zeta )+\Psi (\eta )G(\eta )\), \(\mathcal{Q}(\zeta,\eta )=\Psi (\zeta )G(\eta )+\Psi (\eta )G(\zeta )\), and

$$\begin{aligned} &\Delta _{1}(\xi ):=\frac{1}{m_{1}}\frac{1}{m_{2}} \sum_{p=1}^{m_{1}} \bigl[1-(1-\xi )^{p} \bigr]\sum_{p=1}^{m_{2}} \bigl[1-(1-\xi )^{p} \bigr]; \\ &\Delta _{2}(\xi ):=\frac{1}{m_{1}}\frac{1}{m_{2}}\sum _{p=1}^{m_{1}} \bigl[1-(1-\xi )^{p} \bigr]\sum _{p=1}^{m_{2}} \bigl[1-{\xi }^{p} \bigr]; \\ &\Delta _{3}(\xi ):=\frac{1}{m_{1}}\frac{1}{m_{2}}\sum _{p=1}^{m_{1}} \bigl[1-\xi ^{p} \bigr]\sum _{p=1}^{m_{2}} \bigl[1-(1-\xi )^{p} \bigr]; \\ &\Delta _{4}(\xi ):=\frac{1}{m_{1}}\frac{1}{m_{2}}\sum _{p=1}^{m_{1}} \bigl[1-\xi ^{p} \bigr]\sum _{p=1}^{m_{2}} \bigl[1-{\xi }^{p} \bigr]. \end{aligned}$$

The inclusions are reversed if \(\Psi \in {\mathbf{VP}}_{m_{1}} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\)and \(G\in {\mathbf{VP}}_{m_{2}} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\).

Proof

Let \(\Psi \in {\mathbf{XP}}_{m_{1}} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\) and \(G\in {\mathbf{XP}}_{m_{2}} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\). Then, for \(\xi \in [0,1]\), we have

$$\begin{aligned} \frac{1}{m_{1}}\sum_{p=1}^{m_{1}} \bigl[1-(1-\xi )^{p} \bigr]\Psi ( \zeta )+\frac{1}{m_{1}}\sum _{p=1}^{m_{1}} \bigl[1-{\xi }^{p} \bigr] \Psi (\eta )\subseteq \Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr) \end{aligned}$$

and

$$\begin{aligned} \frac{1}{m_{2}}\sum_{p=1}^{m_{2}} \bigl[1-(1-\xi )^{p} \bigr]G( \zeta )+\frac{1}{m_{2}}\sum _{p=1}^{m_{2}} \bigl[1-{\xi }^{p} \bigr]G( \eta )\subseteq G \bigl(\xi \zeta +(1-\xi )\eta \bigr). \end{aligned}$$

So,

$$\begin{aligned} & \Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr) G \bigl(\xi \zeta +(1-\xi ) \eta \bigr) \\ &\quad\supseteq \frac{1}{m_{1}}\frac{1}{m_{2}}\sum _{p=1}^{m_{1}} \bigl[1-(1- \xi )^{p} \bigr]\sum _{p=1}^{m_{2}} \bigl[1-(1-\xi )^{p} \bigr] \Psi (\zeta )G(\zeta ) \\ & \qquad{}+\frac{1}{m_{1}}\frac{1}{m_{2}}\sum_{p=1}^{m_{1}} \bigl[1-(1- \xi )^{p} \bigr]\sum_{p=1}^{m_{2}} \bigl[1-{\xi }^{p} \bigr]\Psi ( \zeta )G(\eta ) \\ &\qquad{} + \frac{1}{m_{1}}\frac{1}{m_{2}}\sum_{p=1}^{m_{1}} \bigl[1- \xi ^{p} \bigr]\sum_{p=1}^{m_{2}} \bigl[1-(1-\xi )^{p} \bigr]\Psi ( \eta )G(\zeta ) \\ &\qquad{} +\frac{1}{m_{1}}\frac{1}{m_{2}}\sum_{p=1}^{m_{1}} \bigl[1- \xi ^{p} \bigr]\sum_{p=1}^{m_{2}} \bigl[1-{\xi }^{p} \bigr]\Psi ( \eta )G(\eta ) \\ &\quad:=\Delta _{1}(\xi )\Psi (\zeta )G(\zeta )+\Delta _{2}(\xi )\Psi ( \zeta )G(\eta )+ \Delta _{3}(\xi )\Psi (\eta )G(\zeta )+\Delta _{4}( \xi )\Psi (\eta )G(\eta ). \end{aligned}$$

This implies that

$$\begin{aligned} & \Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr) G \bigl( \xi \zeta +(1-\xi ) \eta \bigr) \\ &\quad\supseteq \Delta _{1}(\xi )\Psi (\zeta )G(\zeta )+\Delta _{2}(\xi ) \Psi (\zeta )G(\eta )+ \Delta _{3}(\xi )\Psi ( \eta )G(\zeta )+\Delta _{4}( \xi )\Psi (\eta )G(\eta ). \end{aligned}$$
(24)

Similarly,

$$\begin{aligned} & \Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) G \bigl( \xi \eta +(1-\xi ) \zeta \bigr) \\ &\quad\supseteq \Delta _{4}(\xi )\Psi (\zeta )G(\zeta )+\Delta _{3}(\xi ) \Psi (\zeta )G(\eta )+ \Delta _{2}(\xi )\Psi ( \eta )G(\zeta )+\Delta _{1}( \xi )\Psi (\eta )G(\eta ). \end{aligned}$$
(25)

Adding (24) and (25) gives

$$\begin{aligned} & \Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr) G \bigl( \xi \zeta +(1-\xi ) \eta \bigr)+\Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) G \bigl( \xi \eta +(1- \xi )\zeta \bigr) \\ &\quad\supseteq \bigl(\Psi (\zeta )G(\zeta )+\Psi (\eta )G(\eta ) \bigr) \bigl[ \Delta _{1}(\xi )+\Delta _{4}(\xi ) \bigr] \\ &\qquad{} + \bigl(\Psi (\zeta )G(\eta )+\Psi (\eta )G(\zeta ) \bigr) \bigl[ \Delta _{2}(\xi )+\Delta _{3}(\xi ) \bigr] \\ &\quad:=\mathcal{P}(\zeta,\eta ) \bigl[\Delta _{1}(\xi )+\Delta _{4}(\xi ) \bigr]+\mathcal{Q}(\zeta,\eta ) \bigl[\Delta _{2}(\xi )+\Delta _{3}( \xi ) \bigr]. \end{aligned}$$
(26)

Now, multiplying both sides of (26) by \(\xi ^{\frac{\epsilon }{\rho }-1}\) and integrating the resultant with respect to ξ over \([0,1]\) gives

$$\begin{aligned} &\frac{\rho \Gamma _{\rho }(\epsilon )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )G( \eta )+_{\rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon } \Psi (\zeta )G( \zeta ) \bigr] \\ &\quad= \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \Psi \bigl(\xi \zeta +(1- \xi )\eta \bigr) G \bigl(\xi \zeta +(1-\xi )\eta \bigr) \,d\xi \\ &\qquad{}+ \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \Psi \bigl(\xi \eta +(1- \xi )\zeta \bigr) G \bigl(\xi \eta +(1-\xi )\zeta \bigr) \,d\xi \\ &\quad\supseteq \mathcal{P}(\zeta,\eta ) \int _{0}^{1}\xi ^{ \frac{\epsilon }{\rho }-1} \bigl[\Delta _{1}(\xi )+\Delta _{4}(\xi ) \bigr] \,d\xi +\mathcal{Q}( \zeta,\eta ) \int _{0}^{1}\xi ^{ \frac{\epsilon }{\rho }-1} \bigl[\Delta _{2}(\xi )+\Delta _{3}(\xi ) \bigr] \,d\xi. \end{aligned}$$

Hence, that completes the proof. □

Corollary 13

Let \(\rho, \epsilon >0\). If \(\Psi, G:{\mathbf{S}}\to {\mathbb{K}}_{c}^{+}\)are two convex interval-valued functions with \(\zeta <\eta \), \(\zeta,\eta \in {\mathbf{S}}\)and ΨG is Lebesgue integrable on \([\zeta,\eta ]\), then

$$\begin{aligned} &\frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )G( \eta )+_{\rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon } \Psi (\zeta )G( \zeta ) \bigr] \\ &\quad\supseteq \mathcal{P}(\zeta,\eta ) \biggl[1- \biggl( \frac{2\epsilon }{\epsilon +\rho }- \frac{2\epsilon }{\epsilon +2\rho } \biggr) \biggr]+\mathcal{Q}(\zeta,\eta ) \biggl[ \frac{2\epsilon }{\epsilon +\rho }-\frac{2\epsilon }{\epsilon +2\rho } \biggr]. \end{aligned}$$

Proof

Let \(m_{1}=m_{2}=1\). Then \(\Delta _{1}(\xi )=\xi ^{2}\), \(\Delta _{2}(\xi )= \Delta _{1}(\xi )=\xi -\xi ^{2}\), and \(\Delta _{4}(\xi )=1-2\xi +\xi ^{2}\). We get the desired inequality by applying Theorem 12. □

Remark 14

Corollary 13 boils down to [7, Theorem 3.5] if we set \(\rho =1\).

Theorem 15

Let \(\Psi, G:{\mathbf{S}}\to {\mathbb{K}}_{c}^{+}\)be two interval-valued functions with \(\zeta <\eta \)and \(\zeta,\eta \in {\mathbf{S}}\), and suppose that ΨG is Lebesgue integrable on \([\zeta,\eta ]\). If \(\rho, \epsilon >0\), \(\Psi \in {\mathbf{XP}}_{m_{1}} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\), and \(G\in {\mathbf{XP}}_{m_{2}} ({\mathbf{S}},{\mathbb{K}}_{c}^{+} )\), then

$$\begin{aligned} &\frac{m_{1}m_{2}}{(m_{1}+2^{-m_{1}}-1)(m_{2}+2^{-m_{2}}-1)}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr) G \biggl(\frac{\zeta +\eta }{2} \biggr) \\ &\quad\supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )G( \eta )+_{\rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon } \Psi (\zeta )G( \zeta ) \bigr] \\ & \qquad{}+\frac{\epsilon }{\rho } \int _{0}^{1}\xi ^{ \frac{\epsilon }{\rho }-1} \bigl\{ \bigl[ \Lambda _{m_{1}}(\xi ) \tilde{\Lambda }_{m_{2}}(\xi )+\tilde{\Lambda }_{m_{1}}(\xi )\Lambda _{m_{2}}( \xi ) \bigr] \mathcal{P}(\zeta, \eta ) \\ &\qquad{} + \bigl[\Lambda _{m_{1}}(\xi ) \Lambda _{m_{2}}(\xi )+ \tilde{\Lambda }_{m_{1}}(\xi )\tilde{\Lambda }_{m_{2}}( \xi ) \bigr] \mathcal{Q}(\zeta,\eta ) \bigr\} \,d\xi, \end{aligned}$$

where \(\mathcal{P}(\zeta,\eta )\)and \(\mathcal{Q}(\zeta,\eta )\)are as defined in Theorem 12, and for \(\xi \in [0, 1]\),

$$\begin{aligned} &\Lambda _{m}(\xi )=\frac{1}{m}\sum _{p=1}^{m} \bigl[1-(1-\xi )^{p} \bigr], \\ &\tilde{\Lambda }_{m}(\xi )=\frac{1}{m}\sum _{p=1}^{m} \bigl[1-\xi ^{p} \bigr]. \end{aligned}$$

Proof

First, we observe that from the definitions of \(\tilde{\Lambda }_{m}\) and \(\Lambda _{m}\) given above, we have

$$\begin{aligned} \tilde{\Lambda }_{m} \biggl(\frac{1}{2} \biggr)=\Lambda _{m} \biggl( \frac{1}{2} \biggr):=L_{m}:= \frac{m+2^{-m}-1}{m}. \end{aligned}$$

Hence, from (12), one gets

$$\begin{aligned} L_{m_{1}} \bigl\{ \Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr)+\Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) \bigr\} \subseteq \Psi \biggl( \frac{\zeta +\eta }{2} \biggr) \end{aligned}$$

and

$$\begin{aligned} L_{m_{2}} \bigl\{ G \bigl(\xi \zeta +(1-\xi )\eta \bigr)+G \bigl(\xi \eta +(1- \xi )\zeta \bigr) \bigr\} \subseteq G \biggl(\frac{\zeta +\eta }{2} \biggr). \end{aligned}$$

Now,

$$\begin{aligned} &\Psi \biggl(\frac{\zeta +\eta }{2} \biggr) G \biggl( \frac{\zeta +\eta }{2} \biggr) \\ &\quad\supseteq L_{m_{1}}L_{m_{2}} \bigl[\Psi \bigl(\xi \zeta +(1-\xi ) \eta \bigr) G \bigl(\xi \zeta +(1-\xi )\eta \bigr) \\ &\qquad{} + \Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) G \bigl(\xi \eta +(1- \xi )\zeta \bigr) \bigr] \\ &\qquad{} +L_{m_{1}}L_{m_{2}} \bigl[\Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr) G \bigl(\xi \eta +(1-\xi )\zeta \bigr) \\ & \qquad{}+ \Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) G \bigl(\xi \zeta +(1-\xi )\eta \bigr) \bigr] \\ &\quad\supseteq L_{m_{1}}L_{m_{2}} \bigl[\Psi \bigl(\xi \zeta +(1-\xi ) \eta \bigr) G \bigl(\xi \zeta +(1-\xi )\eta \bigr) \\ & \qquad{}+ \Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) G \bigl(\xi \eta +(1- \xi )\zeta \bigr) \bigr] \\ &\qquad{} + L_{m_{1}}L_{m_{2}} \bigl\{ \bigl[\Lambda _{m_{1}}(\xi ) \Psi ( \zeta )+\tilde{\Lambda }_{m_{1}}(\xi )\Psi (\eta ) \bigr] \bigl[ \Lambda _{m_{2}}(\xi )G(\eta )+\tilde{\Lambda }_{m_{2}}(\xi )G(\zeta ) \bigr] \\ &\qquad{} + \bigl[\Lambda _{m_{1}}(\xi )\Psi ( \eta )+\tilde{\Lambda }_{m_{1}}(\xi )\Psi (\zeta ) \bigr] \bigl[ \Lambda _{m_{2}}(\xi )G( \zeta )+\tilde{\Lambda }_{m_{2}}(\xi )G(\eta ) \bigr] \bigr\} \\ &\quad= L_{m_{1}}L_{m_{2}} \bigl[\Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr) G \bigl(\xi \zeta +(1-\xi )\eta \bigr) \\ &\qquad{} + \Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) G \bigl(\xi \eta +(1- \xi )\zeta \bigr) \bigr] \\ &\qquad{} +L_{m_{1}}L_{m_{2}} \bigl\{ \bigl[\Lambda _{m_{1}}(\xi ) \tilde{\Lambda }_{m_{2}}(\xi )+\tilde{\Lambda }_{m_{1}}(\xi )\Lambda _{m_{2}}( \xi ) \bigr] \bigl(\Psi (\zeta )G(\zeta )+\Psi (\eta )G(\eta ) \bigr) \\ & \qquad{}+ \bigl[\Lambda _{m_{1}}(\xi ) \Lambda _{m_{2}}(\xi )+ \tilde{\Lambda }_{m_{1}}(\xi )\tilde{\Lambda }_{m_{2}}( \xi ) \bigr] \bigl(\Psi (\zeta )G(\eta )+\Psi (\eta )G(\zeta ) \bigr) \bigr\} \\ &\quad:= L_{m_{1}}L_{m_{2}} \bigl[\Psi \bigl(\xi \zeta +(1-\xi )\eta \bigr) G \bigl(\xi \zeta +(1-\xi )\eta \bigr) \\ &\qquad{} + \Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) G \bigl(\xi \eta +(1- \xi )\zeta \bigr) \bigr] \\ &\qquad{} +L_{m_{1}}L_{m_{2}} \bigl\{ \bigl[\Lambda _{m_{1}}(\xi ) \tilde{\Lambda }_{m_{2}}(\xi )+\tilde{\Lambda }_{m_{1}}(\xi )\Lambda _{m_{2}}( \xi ) \bigr] \mathcal{P}(\zeta,\eta ) \\ &\qquad{} + \bigl[\Lambda _{m_{1}}(\xi ) \Lambda _{m_{2}}(\xi )+ \tilde{\Lambda }_{m_{1}}(\xi )\tilde{\Lambda }_{m_{2}}( \xi ) \bigr] \mathcal{Q}(\zeta,\eta ) \bigr\} . \end{aligned}$$

Thus, we get

$$\begin{aligned} &\Psi \biggl(\frac{\zeta +\eta }{2} \biggr) G \biggl( \frac{\zeta +\eta }{2} \biggr) \\ &\quad\supseteq L_{m_{1}}L_{m_{2}} \bigl[\Psi \bigl(\xi \zeta +(1-\xi ) \eta \bigr) G \bigl(\xi \zeta +(1-\xi )\eta \bigr) \\ &\qquad{} + \Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) G \bigl(\xi \eta +(1- \xi )\zeta \bigr) \bigr] \\ &\qquad{} +L_{m_{1}}L_{m_{2}} \bigl\{ \bigl[\Lambda _{m_{1}}(\xi ) \tilde{\Lambda }_{m_{2}}(\xi )+\tilde{\Lambda }_{m_{1}}(\xi )\Lambda _{m_{2}}( \xi ) \bigr] \mathcal{P}(\zeta,\eta ) \\ &\qquad{} + \bigl[\Lambda _{m_{1}}(\xi ) \Lambda _{m_{2}}(\xi )+ \tilde{\Lambda }_{m_{1}}(\xi )\tilde{\Lambda }_{m_{2}}( \xi ) \bigr] \mathcal{Q}(\zeta,\eta ) \bigr\} . \end{aligned}$$
(27)

Multiplying (27) by \(\xi ^{\frac{\epsilon }{\rho }-1}\) and integrating with respect to ξ over \([0,1]\), we get the following inclusion:

$$\begin{aligned} &\frac{\rho }{\epsilon }\Psi \biggl(\frac{\zeta +\eta }{2} \biggr) G \biggl( \frac{\zeta +\eta }{2} \biggr) \\ &\quad= \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1}\Psi \biggl( \frac{\zeta +\eta }{2} \biggr) G \biggl(\frac{\zeta +\eta }{2} \biggr) \,d\xi \\ &\quad\supseteq L_{m_{1}}L_{m_{2}} \int _{0}^{1}\xi ^{ \frac{\epsilon }{\rho }-1} \bigl[\Psi \bigl( \xi \zeta +(1-\xi )\eta \bigr) G \bigl(\xi \zeta +(1-\xi )\eta \bigr) \\ &\qquad{} + \Psi \bigl(\xi \eta +(1-\xi )\zeta \bigr) G \bigl(\xi \eta +(1- \xi )\zeta \bigr) \bigr] \,d\xi \\ &\qquad{} +L_{m_{1}}L_{m_{2}} \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \bigl\{ \bigl[ \Lambda _{m_{1}}(\xi )\tilde{\Lambda }_{m_{2}}(\xi )+ \tilde{\Lambda }_{m_{1}}(\xi )\Lambda _{m_{2}}(\xi ) \bigr] \mathcal{P}(\zeta,\eta ) \\ &\qquad{} + \bigl[\Lambda _{m_{1}}(\xi ) \Lambda _{m_{2}}(\xi )+ \tilde{\Lambda }_{m_{1}}(\xi )\tilde{\Lambda }_{m_{2}}( \xi ) \bigr] \mathcal{Q}(\zeta,\eta ) \bigr\} \,d\xi \\ &\quad=L_{m_{1}}L_{m_{2}} \frac{\rho \Gamma _{\rho }(\epsilon )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{ \mathcal{J}}}_{\zeta ^{+}}^{\epsilon }\Psi (\eta )G( \eta )+_{\rho }{ \mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta )G( \zeta ) \bigr] \\ &\qquad{} +L_{m_{1}}L_{m_{2}} \int _{0}^{1}\xi ^{\frac{\epsilon }{\rho }-1} \bigl\{ \bigl[ \Lambda _{m_{1}}(\xi )\tilde{\Lambda }_{m_{2}}(\xi )+ \tilde{\Lambda }_{m_{1}}(\xi )\Lambda _{m_{2}}(\xi ) \bigr] \mathcal{P}(\zeta,\eta ) \\ &\qquad{} + \bigl[\Lambda _{m_{1}}(\xi ) \Lambda _{m_{2}}(\xi )+ \tilde{\Lambda }_{m_{1}}(\xi )\tilde{\Lambda }_{m_{2}}( \xi ) \bigr] \mathcal{Q}(\zeta,\eta ) \bigr\} \,d\xi, \end{aligned}$$

from which the desired inequality is obtained. □

Corollary 16

Let \(\rho, \epsilon >0\). If \(\Psi, G:{\mathbf{S}}\to {\mathbb{K}}_{c}^{+}\)are two convex interval-valued functions with \(\zeta <\eta \), \(\zeta,\eta \in {\mathbf{S}}\), and ΨG is Lebesgue integrable on \([\zeta,\eta ]\), then

$$\begin{aligned} & \Psi \biggl(\frac{\zeta +\eta }{2} \biggr) G \biggl( \frac{\zeta +\eta }{2} \biggr) \\ &\quad\supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{4(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )G( \eta )+_{\rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon } \Psi (\zeta )G( \zeta ) \bigr] \\ &\qquad{} + \biggl[\frac{\epsilon }{2(\epsilon +\rho )}- \frac{\epsilon }{2(\epsilon +2\rho )} \biggr]\mathcal{P}(\zeta,\eta )+ \biggl[\frac{1}{4}- \biggl(\frac{\epsilon }{2(\epsilon +\rho )}- \frac{\epsilon }{2(\epsilon +2\rho )} \biggr) \biggr]\mathcal{Q}( \zeta,\eta ). \end{aligned}$$

Proof

Let \(m_{1}=m_{2}=1\). Then \(\Lambda _{m_{1}}(\xi )=\Lambda _{m_{2}}(\xi )=\xi \) and \(\tilde{\Lambda }_{m_{1}}(\xi )=\tilde{\Lambda }_{m_{2}}(\xi )=1-\xi \). Using Theorem 12, we get the required result. □

Remark 17

If we take \(\rho =1\), then Corollary 16 becomes [7, Theorem 3.6].

Conclusion

Some new set inclusions of the Hermite–Hadamard types are established for the class of m-polynomial convex interval-valued functions. A relationship between a given m-polynomial convex (concave) interval-valued function \(\Psi =[\psi ^{-},\psi ^{+}]\) and its component real-valued functions \(\psi ^{-}\) and \(\psi ^{+}\) is established. We pointed out some corollaries from which loads of interesting results can be deduced. In addition to these corollaries, if we take \(\psi ^{-}=\psi ^{+}=\psi \), then \(\Psi =\psi \) and the inclusions in Theorems 10, 12, and 15 become the following inequalities:

  1. 1.
    $$\begin{aligned} \frac{m}{m+2^{-m}-1}\psi \biggl(\frac{\zeta +\eta }{2} \biggr)&\leq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\psi (\zeta ) \bigr] \\ & \leq \frac{\psi (\zeta )+\psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ); \end{aligned}$$
  2. 2.
    $$\begin{aligned} &\frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \psi (\eta )g( \eta )+_{\rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon } \psi (\zeta )g( \zeta ) \bigr] \\ &\quad\leq \frac{\epsilon \mathcal{P}(\zeta,\eta )}{\rho } \int _{0}^{1} \xi ^{\frac{\epsilon }{\rho }-1} \bigl[\Delta _{1}(\xi )+\Delta _{4}( \xi ) \bigr] \,d\xi \\ &\qquad{} + \frac{\epsilon \mathcal{Q}(\zeta,\eta )}{\rho } \int _{0}^{1}\xi ^{ \frac{\epsilon }{\rho }-1} \bigl[\Delta _{2}(\xi )+\Delta _{3}(\xi ) \bigr] \,d\xi; \end{aligned}$$

    and

  3. 3.
    $$\begin{aligned} &\frac{m_{1}m_{2}}{(m_{1}+2^{-m_{1}}-1)(m_{2}+2^{-m_{2}}-1)}\psi \biggl(\frac{\zeta +\eta }{2} \biggr) g \biggl(\frac{\zeta +\eta }{2} \biggr) \\ &\quad\leq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \psi (\eta )g( \eta )+_{\rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon } \psi (\zeta )g( \zeta ) \bigr] \\ &\qquad{} +\frac{\epsilon }{\rho } \int _{0}^{1}\xi ^{ \frac{\epsilon }{\rho }-1} \bigl\{ \bigl[ \Lambda _{m_{1}}(\xi ) \tilde{\Lambda }_{m_{2}}(\xi )+\tilde{\Lambda }_{m_{1}}(\xi )\Lambda _{m_{2}}( \xi ) \bigr] \mathcal{P}(\zeta, \eta ) \\ &\qquad{} + \bigl[\Lambda _{m_{1}}(\xi ) \Lambda _{m_{2}}(\xi )+ \tilde{\Lambda }_{m_{1}}(\xi )\tilde{\Lambda }_{m_{2}}( \xi ) \bigr] \mathcal{Q}(\zeta,\eta ) \bigr\} \,d\xi, \end{aligned}$$

respectively.

References

  1. 1.

    Adil Khan, M., Ali, T., Khan, T.U.: Hermite–Hadamard type inequalities with applications. Fasc. Math. 59, 57–74 (2017)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Adil Khan, M., Iqbal, A., Suleman, M., Chu, Y.-M.: Hermite–Hadamard type inequalities for fractional integrals via Green function. J. Inequal. Appl. 2018, Article ID 161 (2018)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Adil Khan, M., Khurshid, Y., Ali, T.: Hermite–Hadamard inequality for fractional integrals via η-convex functions. Acta Math. Univ. Comen. 86(1), 153–164 (2017)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Adil Khan, M., Khurshid, Y., Du, T., Chu, Y.-M.: Generalization of Hermite–Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces 2018, Article ID 5357463 (2018)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Adil Khan, M., Mohammad, N., Nwaeze, E.R., Chu, Y.-M.: Quantum Hermite–Hadamard inequality by means of a Green function. Adv. Differ. Equ. 2020, 99 (2020)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Breckner, W.W.: Continuity of generalized convex and generalized concave set-valued functions. Rev. Anal. Numér. Théor. Approx. 22, 39–51 (1993)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Budak, H., Tunç, T., Sarikaya, M.Z.: Fractional Hermite–Hadamard-type inequalities for interval-valued functions. Proc. Am. Math. Soc. 148(2), 705–718 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Chalco-Cano, Y., Flores-Franulic̆, A., Román-Flores, H.: Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative. Comput. Appl. Math. 31, 457–472 (2012)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Chalco-Cano, Y., Lodwick, W.A., Condori-Equice, W.: Ostrowski type inequalities and applications in numerical integration for interval-valued functions. Soft Comput. 19, 3293–3300 (2015)

    MATH  Article  Google Scholar 

  10. 10.

    Chu, Y.-M., Adil Khan, M., Khan, T.U., Ali, T.: Generalizations of Hermite–Hadamard type inequalities for MT-convex functions. J. Nonlinear Sci. Appl. 9, 4305–4316 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Chu, Y.-M., Adil Khan, M., Khan, T.U., Khan, J.: Some new inequalities of Hermite–Hadamard type for s-convex functions with applications. Open Math. 15(1), 1414–1430 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Costa, T.M.: Jensen’s inequality type integral for fuzzy-interval-valued functions. Fuzzy Sets Syst. 327, 31–47 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Costa, T.M., Román-Flores, H.: Some integral inequalities for fuzzy-interval-valued functions. Inf. Sci. 420, 110–125 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Delavar, M.R., De La Sen, M.: Some generalizations of Hermite–Hadamard type inequalities. SpringerPlus 5, 1661 (2016)

    Article  Google Scholar 

  15. 15.

    Guessab, A., Schmeisser, G.: Sharp integral inequalities of the Hermite–Hadamard type. J. Approx. Theory 115(2), 260–288 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Hadamard, J.: Étude sur les propriétés des fonctions entiéres et en particulier d’une fonction considerée par Riemann. J. Math. Pures Appl. 58, 171–215 (1893)

    MATH  Google Scholar 

  17. 17.

    Hermite, C.: Sur deux limites d’une intégrale dé finie. Mathesis 3, 82 (1883)

    Google Scholar 

  18. 18.

    Iqbal, A., Adil Khan, M., Mohammad, N., Nwaeze, E.R., Chu, Y.-M.: Revisiting the Hermite–Hadamard fractional integral inequality via a Green function. AIMS Math. 5(6), 6087–6107 (2020)

    Article  Google Scholar 

  19. 19.

    Iqbal, A., Adil Khan, M., Suleman, M., Chu, Y.-M.: The right Riemann–Liouville fractional Hermite–Hadamard type inequalities derived from Green’s function. AIP Adv. 10, Article ID 045032 (2020)

    Article  Google Scholar 

  20. 20.

    Iqbal, A., Adil Khan, M., Ullah, S., Chu, Y.-M.: Some new Hermite–Hadamard type inequalities associated with conformable fractional integrals and their applications. J. Funct. Spaces 2020, Article ID 9845407 (2020)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Iqbal, A., Adil Khan, M., Ullah, S., Kashuri, A., Chu, Y.-M.: Hermite–Hadamard type inequalities pertaining conformable fractional integrals and their applications. AIP Adv. 8, Article ID 075101 (2018)

    Article  Google Scholar 

  22. 22.

    Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Hermite–Hadamard–Fejer inequalities for conformable fractional integrals via preinvex functions. J. Funct. Spaces 2019, Article ID 3146210 (2019)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Ostrowski type inequalities involving conformable integrals via preinvex functions. AIP Adv. 10, Article ID 055204 (2020)

    Article  Google Scholar 

  24. 24.

    Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Generalized inequalities via GG-convexity and GA-convexity. AIMS Math. 5(5), 5012–5030 (2020)

    Article  Google Scholar 

  25. 25.

    Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Conformable integral version of Hermite–Hadamard–Fejér inequalities via η-convex functions. AIMS Math. 5(5), 5106–5120 (2020)

    Article  Google Scholar 

  26. 26.

    Khurshid, Y., Adil Khan, M., Ming Chu, Y.: Generalized inequalities via GG-convexity and GA-convexity. J. Funct. Spaces 2019, Article ID 6926107 (2019)

    MATH  Google Scholar 

  27. 27.

    Matkowski, J., Nikodem, K.: An integral Jensen inequality for convex multifunctions. Results Math. 26, 348–353 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Mitroi, F.-C., Nikodem, K., Wąsowicz, S.: Hermite–Hadamard inequalities for convex set-valued functions. Demonstr. Math. 46, 655–662 (2013)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  30. 30.

    Moore, R.E.: Method and Applications of Interval Analysis. SIAM, Philadelphia (1979)

    Book  Google Scholar 

  31. 31.

    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)

    MATH  Book  Google Scholar 

  32. 32.

    Nikodem, K., Sánchez, J.L., Sánchez, L.: Jensen and Hermite–Hadamard inequalities for strongly convex set-valued maps. Math. Æterna 4, 979–987 (2014)

    Google Scholar 

  33. 33.

    Nwaeze, E.R.: Inequalities of the Hermite–Hadamard type for quasi-convex functions via the \((k,s)\)-Riemann–Liouville fractional integrals. Fract. Differ. Calc. 8(2), 327–336 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Nwaeze, E.R., Torres, D.F.M.: Novel results on the Hermite–Hadamard kind inequality for η-convex functions by means of the \((k,r)\)-fractional integral operators. In: Sever Dragomir, S., Agarwal, P., Jleli, M., Samet, B. (eds.) Advances in Mathematical Inequalities and Applications (AMIA). Trends in Mathematics, pp. 311–321. Birkhäuser, Singapore (2018)

    Chapter  Google Scholar 

  35. 35.

    Román-Flores, H., Chalco-Cano, Y., Lodwick, W.A.: Some integral inequalities for interval-valued functions. Comput. Appl. Math. 35, 1–13 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Sadowska, E.: Hadamard inequality and a refinement of Jensen inequality for set valued functions. Results Math. 32, 332–337 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    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)

    MATH  Article  Google Scholar 

  38. 38.

    Sun, J., Xi, B.-Y., Qi, F.: Some new inequalities of the Hermite–Hadamard type for extended s-convex functions. J. Comput. Anal. Appl. 26(6), 985–996 (2019)

    Google Scholar 

  39. 39.

    Toplu, T., Kadakal, M., Íşcan, Í.: On n-polynomial convexity and some related inequalities. AIMS Math. 5(2), 1304–1318 (2020)

    Article  Google Scholar 

  40. 40.

    Zhao, D., An, T., Ye, G., Liu, W.: New Jensen and Hermite–Hadamard type inequalities for h-convex interval-valued functions. J. Inequal. Appl. 2018, 302 (2018)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Zhao, D., An, T., Ye, G., Torres, D.F.M.: On Hermite–Hadamard type inequalities for harmonical h-convex interval-valued functions. Math. Inequal. Appl. 23(1), 95–105 (2020)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Many thanks to the referees for their valuable comments and suggestions.

Availability of data and materials

Not applicable.

Funding

This work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485).

Author information

Affiliations

Authors

Contributions

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

Corresponding author

Correspondence to Yu-Ming Chu.

Ethics declarations

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nwaeze, E.R., Khan, M.A. & Chu, YM. Fractional inclusions of the Hermite–Hadamard type for m-polynomial convex interval-valued functions. Adv Differ Equ 2020, 507 (2020). https://doi.org/10.1186/s13662-020-02977-3

Download citation

MSC

  • 26D15
  • 26E25
  • 28B20

Keywords

  • Hermite–Hadamard
  • m-polynomial convex
  • Interval-valued function
  • ρ-Riemann–Liouville