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Some fractional Hermite–Hadamard-type inequalities for interval-valued coordinated functions

Abstract

The primary objective of this paper is establishing new Hermite–Hadamard-type inequalities for interval-valued coordinated functions via Riemann–Liouville-type fractional integrals. Moreover, we obtain some fractional Hermite–Hadamard-type inequalities for the product of two coordinated h-convex interval-valued functions. Our results generalize several well-known inequalities.

Introduction

The classical Hermite–Hadamard inequalities state that

$$\begin{aligned} f \biggl(\frac{o+\varsigma }{2} \biggr)\leq \frac{1}{\varsigma -o} \int _{o}^{ \varsigma }f(\chi )\,d\chi \leq \frac{f(o)+f(\varsigma )}{2}, \end{aligned}$$
(1.1)

where \(f:\mathcal{I}\rightarrow \mathbb{R}\) is a convex function on the closed bounded interval \(\mathcal{I}\) of \(\mathbb{R}\), and \(o,\varsigma \in \mathcal{I}\) with \(o<\varsigma \). Since they play a crucial role in convex analysis and can be a very powerful tool for measuring and computing errors, many authors have devoted their efforts to generalize inequalities (1.1); see [16]. It is worth noting that Sarikaya et al. [7] established new Hermite–Hadamard-type inequalities via the Riemann–Liouville fractional integrals. Since then, many papers have generalized different forms of fractional integrals and presented new and interesting refinements of Hermite–Hadamard-type inequalities using these integrals. Fernandez and Mohammed [8] established some Hermite–Hadamard-type inequalities for the Atangana–Baleanu fractional integral. Mohammed and Abdeljawad [9] proved new Hermite–Hadamard-type inequalities in the context of fractional calculus with respect to functions involving nonsingular kernels. For other related results, we refer the readers to [719].

On the other hand, to improve the reliability of the calculation results and automatic operation error analysis, Moore [20] introduced the theory of interval analysis. Interval analysis has a strong model for handling interval uncertainty and has been widely applied and stretched in control theory [21], dynamical game theory [22], and many others. Recently, numerous famous inequalities have been extended to interval-valued functions. Chalco-Cano et al. [23] obtained Ostrowski-type inequalities for interval-valued functions by using the Hukuhara derivative. Román-Flores et al. [24] derived the Minkowski and Beckenbach-type inequalities for interval-valued functions. Liu et al. [18] proved Hermite–Hadamard-type inequalities via interval Riemann–Liouville-type fractional integrals for interval-valued functions. Very recently, Zhao et al. [25, 26] established Hermite–Hadamard-type inequalities for interval-valued coordinated functions. Budak et al. [27] gave a definition of Riemann–Liouville-type fractional integrals for interval-valued coordinated functions and presented some new Hermite–Hadamard-type inequalities.

Motivated by Zhao et al. [25, 26] and Budak et al. [27], we present a new class of Hermite–Hadamard-type inequalities for coordinated h-convex interval-valued functions via Riemann–Liouville-type fractional integrals. We also establish Hermite–Hadamard-type inequalities for the products of two interval-valued coordinated functions.

The paper is organized as follows. Section 2 contains some necessary preliminaries. In Sect. 3, we establish some new Hermite–Hadamard-type inequalities for coordinated h-convex interval-valued functions via Riemann–Liouville-type fractional integrals. We end with Sect. 4 of conclusions.

Preliminaries

In this section, we recall some basic definitions and results on interval analysis. We denote by \(\mathbb{R}_{\mathcal{I}}\) the set of closed bounded intervals of \(\mathbb{R}\) and by \(\mathbb{R}^{+}\) and \(\mathbb{R}_{ \mathcal{I}}^{+}\) the sets of positive real numbers and positive intervals, respectively. We also denote \(\triangle =[o,\varsigma ]\times [\rho ,q]\). For more notions on interval-valued functions, see [28, 29].

Definition 2.1

([29])

Let \(h:[0,1]\rightarrow \mathbb{R}^{+}\). We say that \(f:[o,\varsigma ]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}\) is an h-convex interval-valued function if for all \(\chi ,\gamma \in [o,\varsigma ]\) and \(\tau \in [0,1]\), we have

$$\begin{aligned} f\bigl(\tau \chi +(1-\tau )\gamma \bigr)\supseteq h(\tau )f(\chi )+h(1-\tau )f( \gamma ). \end{aligned}$$

We denote the set of all h-convex interval-valued functions by \(SX(h,[o,\varsigma ],\mathbb{R}_{\mathcal{I}}^{+})\).

Definition 2.2

([26])

A function \(\mathcal{F}:\triangle \rightarrow \mathbb{R}_{\mathcal{I}}^{+}\) is said to be a coordinated convex interval-valued function if

$$\begin{aligned} \begin{aligned} &\mathcal{F}\bigl(\tau \chi +(1-\tau )\gamma ,\theta \mu +(1-\theta )\nu \bigr) \\ &\quad \supseteq \tau \theta \mathcal{F}(\chi ,\mu )+\tau (1-\theta ) \mathcal{F}(\chi ,\nu )+(1-\tau )\theta \mathcal{F}(\gamma ,\mu )+(1- \tau ) (1- \theta )\mathcal{F}(\gamma ,\nu ) \end{aligned} \end{aligned}$$

for all \((\chi ,\gamma ),(\mu ,\nu )\in \triangle \) and \(\tau ,\theta \in [0,1]\).

Definition 2.3

([25])

Let \(h:[0,1]\rightarrow \mathbb{R}^{+}\). Then \(\mathcal{F}:\triangle \rightarrow \mathbb{R}_{\mathcal{I}}^{+}\) is called a coordinated h-convex interval-valued function on if the partial mappings

$$\begin{aligned}& \mathcal{F}_{\gamma }:[o,\varsigma ]\rightarrow \mathbb{R}_{ \mathcal{I}}^{+},\qquad \mathcal{F}_{\gamma }(\chi )=\mathcal{F}(\chi , \gamma ), \\& \mathcal{F}_{\chi }:[\rho ,q]\rightarrow \mathbb{R}_{\mathcal{I}}^{+},\qquad \mathcal{F}_{\chi }(\gamma )=\mathcal{F}(\chi ,\gamma ) \end{aligned}$$

are h-convex for all \(\gamma \in [\rho ,q]\) and \(\chi \in [o,\varsigma ]\). We denote the set of all coordinated h-convex interval-valued functions on by \(SX(ch,\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\).

The families of all Riemann-integrable real-valued functions on \([o,\varsigma ]\), interval-valued functions on \([o,\varsigma ]\) and on are denoted by \(\mathcal{R}_{([o,\varsigma ])}\), \(\mathcal{IR}_{([o,\varsigma ])}\), and \(\mathcal{ID}_{(\triangle )}\). We have the following:

Theorem 2.4

([30])

Let \(f:[o,\varsigma ]\rightarrow \mathbb{R}_{\mathcal{I}}\) be such that \(f=[\underline{f},\overline{f}]\). Then \(f\in \mathcal{IR}_{([o,\varsigma ])}\) iff \(\underline{f}\), \(\overline{f}\in \mathcal{R}_{([o,\varsigma ])}\) and

$$ (\mathcal{IR}) \int _{o}^{\varsigma }f(s)\,ds= \biggl[(\mathcal{R}) \int _{o}^{ \varsigma }\underline{f}(s)\,ds,(\mathcal{R}) \int _{o}^{\varsigma } \overline{f}(s)\,ds \biggr]. $$

Theorem 2.5

([31])

Let \(\mathcal{F}:\triangle \rightarrow \mathbb{R}_{\mathcal{I}}\). If \(\mathcal{F}\in \mathcal{ID}_{(\triangle )}\), then

$$ (\mathcal{ID}) \iint _{\triangle } {\mathcal{F}}(t,s)\,dt \,ds=(\mathcal{IR}) \int _{o}^{\varsigma }\,dt(\mathcal{IR}) \int _{\rho }^{q}\mathcal{{F}}(t,s)\,ds. $$

Definition 2.6

([16])

Let \(f:[o,\varsigma ]\rightarrow \mathbb{R}_{\mathcal{I}}\) and \(f\in \mathcal{IR}_{([o,\varsigma ])}\). Then the interval Riemann–Liouville-type integrals of f are defined by

$$ \begin{aligned} &\mathfrak{J}_{o^{+}}^{\alpha }f(\vartheta )=\frac{1}{\Gamma (\alpha )} \int _{o}^{\vartheta }(\vartheta -\chi )^{\alpha -1}f(\chi )\,d\chi ,\quad \vartheta >o, \\ &\mathfrak{J}_{\varsigma ^{-}}^{\alpha }f(\vartheta )= \frac{1}{\Gamma (\alpha )} \int _{\vartheta }^{\varsigma }(\chi - \vartheta )^{\alpha -1}f(\chi )\,d\chi , \quad \vartheta < \varsigma , \end{aligned} $$

where \(\alpha >0\), and Γ is the gamma function.

Theorem 2.7

([32])

Let \(f:[o,\varsigma ]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}\), \(f \in \mathcal{IR}_{([o,\varsigma ])}\), and \(h:[0,1]\rightarrow \mathbb{R}^{+}\). If \(f\in SX(h,[o,\varsigma ], \mathbb{R}^{+}_{\mathcal{I}})\), then

$$\begin{aligned} \begin{aligned} \frac{1}{\alpha h(\frac{1}{2})}{f} \biggl( \frac{o+\varsigma }{2} \biggr) &\supseteq \frac{\Gamma (\alpha )}{(\varsigma -o)^{\alpha }} \bigl[\mathfrak{J}^{\alpha }_{o^{+}}{f}( \varsigma )+ {\mathfrak{J}}^{ \alpha }_{\varsigma ^{-}}{f}(o) \bigr] \\ & \supseteq \bigl[f(o)+f(\varsigma )\bigr]{ \int _{0}^{1} \tau ^{\alpha -1} \bigl[h(\tau )+h(1-\tau ) \bigr]\,d\tau } \end{aligned} \end{aligned}$$
(2.1)

with \(\alpha >0\).

The Riemann–Liouville-type fractional integrals of interval-valued coordinated functions \(\mathcal{F}(t,s)\) are given as follows.

Definition 2.8

([27])

Let \(\mathcal{F}:\triangle \rightarrow \mathbb{R}_{\mathcal{I}}^{+}\) and \(\mathcal{F}\in \mathcal{ID}_{(\triangle )}\). The Riemann–Liouville-type integrals \(\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta }\), \(\mathfrak{J}_{o^{+},q^{-}}^{\alpha ,\beta }\), \(\mathfrak{J}_{ \varsigma ^{-},\rho ^{+}}^{\alpha ,\beta }\), \(\mathfrak{J}_{\varsigma ^{-},q^{-}}^{ \alpha ,\beta }\) of \(\mathcal{F}\) of order \(\alpha ,\beta >0\) are defined by

$$\begin{aligned} &\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta } \mathcal{F}(\chi , \gamma )=\frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{o}^{\chi } \int _{\rho }^{\gamma }(\chi -t)^{\alpha -1}(\gamma -s)^{\beta -1} \mathcal{F}(t,s)\,ds \,dt, \quad \chi >o,\gamma >\rho , \\ &\mathfrak{J}_{o^{+},q^{-}}^{\alpha ,\beta }\mathcal{F}(\chi ,\gamma )= \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{o}^{\chi } \int _{ \gamma }^{q}(\chi -t)^{\alpha -1}(s-\gamma )^{\beta -1}\mathcal{F}(t,s)\,ds \,dt, \quad \chi >o,\gamma < q, \\ &\mathfrak{J}_{\varsigma ^{-},\rho ^{+}}^{\alpha ,\beta }\mathcal{F}( \chi ,\gamma )= \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\chi }^{ \varsigma } \int _{\rho }^{\gamma }(t-\chi )^{\alpha -1}(\gamma -s)^{ \beta -1}\mathcal{F}(t,s)\,ds \,dt, \quad \chi < \varsigma ,\gamma >\rho , \\ &\mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha ,\beta }\mathcal{F}(\chi , \gamma )= \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\chi }^{ \varsigma } \int _{\gamma }^{q}(t-\chi )^{\alpha -1}(s-\gamma )^{\beta -1} \mathcal{F}(t,s)\,ds \,dt, \quad \chi < \varsigma ,\gamma < q. \end{aligned}$$

Main results

In this section, we prove some new Hermite–Hadamard-type inequalities for coordinated h-convex interval-valued functions via the interval Riemann–Liouville-type integrals.

Theorem 3.1

Let \(\mathcal{F}:\triangle \rightarrow \mathbb{R}_{\mathcal{I}}^{+}\) be such that \(\mathcal{F}=[\underline{\mathcal{F}},\overline{\mathcal{F}}]\) and \(\mathcal{F}\in \mathcal{ID}_{(\triangle )}\), and let \(h:[0,1]\rightarrow \mathbb{R}^{+}\). If \(\mathcal{F}\in SX(ch,\triangle ,\mathbb{R}^{+}_{\mathcal{I}})\), then

$$\begin{aligned} \begin{aligned} &\frac{1}{\alpha \beta h^{2}(\frac{1}{2})} \mathcal{F} \biggl( \frac{o+\varsigma }{2},\frac{\rho +q}{2} \biggr) \\ &\quad \supseteq \frac{\Gamma (\alpha )\Gamma (\beta )}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }}\\ &\qquad {}\times \bigl[\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta } \mathcal{F}( \varsigma ,q)+\mathfrak{J}_{o^{+},q^{-}}^{\alpha ,\beta }\mathcal{F}( \varsigma ,\rho )+\mathfrak{J}_{\varsigma ^{-},\rho ^{+}}^{\alpha , \beta }\mathcal{F}(o,q)+ \mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha , \beta }\mathcal{F}(o,\rho ) \bigr] \\ &\quad \supseteq \bigl[\mathcal{F}(o,\rho )+\mathcal{F}(o,q)+\mathcal{F}( \varsigma ,\rho )+\mathcal{F}(\varsigma ,q) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h(\tau )h(\theta )+h(1-\tau )h(\theta )\\ &\qquad {}+h(\tau )h(1-\theta )+h(1- \tau )h(1-\theta ) \bigr]\,d\tau \,d\theta \end{aligned} \end{aligned}$$
(3.1)

with \(\alpha ,\beta >0\).

Proof

Since \(\mathcal{F}\in SX(ch,\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\), we have

$$ \frac{1}{h^{2}(\frac{1}{2})}\mathcal{F} \biggl(\frac{\chi +\gamma }{2}, \frac{\mu +\nu }{2} \biggr)\supseteq \mathcal{F}(\chi ,\mu )+\mathcal{F}( \gamma ,\mu )+\mathcal{F}( \chi ,\nu )+\mathcal{F}(\gamma ,\nu ). $$

Let \(\chi =\tau o+(1-\tau )\varsigma \), \(\gamma =(1-\tau )o+\tau \varsigma \), \(\mu =\theta \rho +(1-\theta )q\), \(w=(1-\theta )\rho +\theta q\), \(\tau ,\theta \in [0,1]\). Then

$$ \begin{aligned} &\frac{1}{h^{2}(\frac{1}{2})} \mathcal{F} \biggl(\frac{o+\varsigma }{2}, \frac{\rho +q}{2} \biggr) \\ &\quad \supseteq \mathcal{F}\bigl(\tau o+(1-\tau ) \varsigma ,\theta \rho +(1- \theta ) q\bigr)+ \mathcal{F}\bigl((1-\tau ) o+\tau \varsigma ,\theta \rho +(1- \theta ) q\bigr) \\ &\qquad {}+\mathcal{F}\bigl(\tau o+(1-\tau ) \varsigma ,(1-\theta )\rho +\theta q\bigr)+ \mathcal{F}\bigl((1-\tau ) o+\tau \varsigma ,(1-\theta ) \rho +\theta q \bigr). \end{aligned} $$
(3.2)

Consequently,

$$\begin{aligned} &\frac{1}{\alpha \beta h^{2}(\frac{1}{2})}\mathcal{F} \biggl( \frac{o+\varsigma }{2},\frac{\rho +q}{2} \biggr) \\ &\quad =\frac{1}{ h^{2}(\frac{1}{2})} \mathcal{F} \biggl( \frac{o+\varsigma }{2}, \frac{\rho +q}{2} \biggr) \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1}\,d\theta \,d\tau \\ &\quad \supseteq \biggl[ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{ \beta -1}\mathcal{F}\bigl(\tau o+(1-\tau ) \varsigma ,\theta \rho +(1- \theta ) q\bigr)\,d\theta \,d\tau \\ &\qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \mathcal{F}\bigl((1-\tau ) o+\tau \varsigma ,\theta \rho +(1- \theta ) q\bigr)\,d \theta \,d\tau \\ & \qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{ \beta -1}\mathcal{F}\bigl(\tau o+(1-\tau ) \varsigma ,(1-\theta ) \rho + \theta q\bigr)\,d\theta \,d\tau \\ &\qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{ \beta -1}\mathcal{F}\bigl((1-\tau ) o+\tau \varsigma ,(1-\theta ) \rho + \theta q\bigr)\,d\theta \,d\tau \biggr] \\ &\quad = \biggl[ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \bigl[\underline{\mathcal{F}}\bigl(\tau o+(1-\tau ) \varsigma , \theta \rho +(1-\theta ) q\bigr), \\ &\qquad \overline{\mathcal{F}}\bigl(\tau o+(1-\tau ) \varsigma ,\theta \rho +(1-\theta ) \,dq\bigr) \bigr]\,d\theta \,d\tau \\ & \qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \bigl[\underline{\mathcal{F}}\bigl((1-\tau ) o+\tau \varsigma , \theta \rho +(1-\theta ) q\bigr), \\ &\qquad\overline{\mathcal{F}}\bigl((1-\tau ) o+\tau \varsigma ,\theta \rho +(1-\theta ) q\bigr) \bigr]\,d\theta \,d\tau \\ &\qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \bigl[\underline{\mathcal{F}}\bigl(\tau o+(1-\tau ) \varsigma ,(1- \theta ) \rho +\theta q\bigr), \\ &\qquad\overline{\mathcal{F}}\bigl(\tau o+(1-\tau ) \varsigma ,(1-\theta ) \rho +\theta q\bigr) \bigr]\,d\theta \,d\tau \\ &\qquad{} + \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \bigl[\underline{\mathcal{F}}\bigl((1-\tau ) o+\tau \varsigma ,(1- \theta ) \rho +\theta q\bigr), \\ &\qquad\overline{\mathcal{F}}\bigl((1-\tau ) o+\tau \varsigma ,(1-\theta ) \rho +\theta q\bigr) \bigr]\,d\theta \,d\tau \biggr] \\ &\quad = \biggl[ \int _{\varsigma }^{o} \int _{q}^{\rho } \bigl(\eta (\chi ) \bigr)^{\alpha -1} \bigl(\zeta (\gamma ) \bigr)^{\beta -1} \underline{ \mathcal{F}}(\chi ,\gamma )\frac{d\gamma }{\rho -q} \frac{d\chi }{o-\varsigma }, \\ &\qquad \int _{\varsigma }^{o} \int _{q}^{\rho } \bigl(\eta (\chi ) \bigr)^{\alpha -1} \bigl(\zeta (\gamma ) \bigr)^{\beta -1} \overline{ \mathcal{F}}(\chi ,\gamma )\frac{d\gamma }{\rho -q} \frac{d\chi }{o-\varsigma } \biggr] \\ &\qquad {}+ \biggl[ \int _{o}^{\varsigma } \int _{q}^{\rho } \bigl(1-\eta (\chi ) \bigr)^{\alpha -1} \bigl(\zeta (\gamma ) \bigr)^{\beta -1} \underline{ \mathcal{F}}(\chi ,\gamma )\frac{d\gamma }{\rho -q} \frac{d\chi }{\varsigma -o}, \\ &\qquad \int _{o}^{\varsigma } \int _{q}^{\rho } \bigl(1-\eta ( \chi ) \bigr)^{\alpha -1} \bigl(\zeta (\gamma ) \bigr)^{\beta -1} \overline{ \mathcal{F}}(\chi ,\gamma )\frac{d\gamma }{\rho -q} \frac{d\chi }{\varsigma -a} \biggr] \\ &\qquad {}+ \biggl[ \int _{\varsigma }^{o} \int _{\rho }^{q} \bigl(\eta (\chi ) \bigr)^{\alpha -1} \bigl(1-\zeta (\gamma ) \bigr)^{\beta -1} \underline{ \mathcal{F}}(\chi ,\gamma )\frac{d\gamma }{q-\rho } \frac{d\chi }{o-\varsigma }, \\ &\qquad \int _{\varsigma }^{o} \int _{\rho }^{q} \bigl(\eta ( \chi ) \bigr)^{\alpha -1} \bigl(1-\zeta (\gamma ) \bigr)^{\beta -1} \overline{ \mathcal{F}}(\chi ,\gamma )\frac{d\gamma }{q-\rho } \frac{d\chi }{o-\varsigma } \biggr] \\ &\qquad {}+ \biggl[ \int _{o}^{\varsigma } \int _{\rho }^{q} \bigl(1-\eta (\chi ) \bigr)^{\alpha -1} \bigl(1-\zeta (\gamma ) \bigr)^{\beta -1} \underline{ \mathcal{F}}(\chi ,\gamma )\frac{d\gamma }{q-\rho } \frac{d\chi }{\varsigma -o}, \\ &\qquad \int _{o}^{\varsigma } \int _{\rho }^{q} \bigl(1-\eta ( \chi ) \bigr)^{\alpha -1} \bigl(1-\zeta (\gamma ) \bigr)^{\beta -1} \overline{ \mathcal{F}}(\chi ,\gamma )\frac{d\gamma }{q-\rho } \frac{d\chi }{\varsigma -o} \biggr] \\ &\quad = \frac{\Gamma (\alpha )\Gamma (\beta )}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \bigl[\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta } \underline{\mathcal{F}}(\varsigma ,q)+\mathfrak{J}_{o^{+},q^{-}}^{ \alpha ,\beta } \underline{\mathcal{F}}(\varsigma ,\rho )+\mathfrak{J}_{ \varsigma ^{-},\rho ^{+}}^{\alpha ,\beta } \underline{\mathcal{F}}(o,q)+ \mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha ,\beta } \underline{\mathcal{F}}(o,\rho ), \\ & \qquad \mathfrak{J}_{o^{+},\rho ^{+}}^{ \alpha ,\beta }\overline{ \mathcal{F}}(\varsigma ,q)+\mathfrak{J}_{o^{+},q^{-}}^{ \alpha ,\beta }\overline{ \mathcal{F}}(\varsigma ,\rho )+\mathfrak{J}_{ \varsigma ^{-},\rho ^{+}}^{\alpha ,\beta } \overline{\mathcal{F}}(o,q)+ \mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha ,\beta } \overline{F}(o, \rho ) \bigr] \\ &\quad =\frac{\Gamma (\alpha )\Gamma (\beta )}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \bigl[\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta } \mathcal{F}( \varsigma ,q)+\mathfrak{J}_{o^{+},q^{-}}^{\alpha ,\beta }\mathcal{F}( \varsigma ,\rho )+\mathfrak{J}_{\varsigma ^{-},\rho ^{+}}^{\alpha , \beta }\mathcal{F}(o,q)+ \mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha , \beta }\mathcal{F}(o,\rho ) \bigr], \end{aligned}$$
(3.3)

where \(\eta (\chi )=\frac{\varsigma -\chi }{\varsigma -o}\), \(\zeta (\gamma )= \frac{q-\gamma }{q-\rho }\).

Similarly, since \(\mathcal{F}\in SX(ch,\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\),

$$ \begin{aligned} &\mathcal{F}\bigl(\tau o+(1-\tau ) \varsigma ,\theta \rho +(1-\theta ) q\bigr)+ \mathcal{F}\bigl((1-\tau ) o+\tau \varsigma ,\theta \rho +(1-\theta ) q\bigr) \\ &\qquad {}+\mathcal{F}\bigl(\tau o+(1-\tau ) \varsigma ,(1-\theta )\rho +\theta q\bigr)+ \mathcal{F}\bigl((1-\tau ) o+\tau \varsigma ,(1-\theta ) \rho +\theta q\bigr) \\ &\quad \supseteq \bigl[h(\tau )h(\theta )+h(1-\tau )h(\theta )+h(\tau )h(1- \theta )+h(1-\tau )h(1-\theta ) \bigr] \\ &\qquad {}\times \bigl[\mathcal{F}(o,\rho )+\mathcal{F}(o,q)+\mathcal{F}( \varsigma ,\rho )+\mathcal{F}(\varsigma ,q) \bigr]. \end{aligned} $$
(3.4)

Multiplying both sides of (3.4) by \(\tau ^{\alpha -1}\theta ^{\beta -1}\) and integrating on \([0,1]\times [0,1]\), we have

$$\begin{aligned} \begin{aligned} &\frac{\Gamma (\alpha )\Gamma (\beta )}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \bigl[ \mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta }\mathcal{F}( \varsigma ,q)+ \mathfrak{J}_{o^{+},q^{-}}^{\alpha ,\beta }\mathcal{F}( \varsigma ,\rho )+ \mathfrak{J}_{\varsigma ^{-},\rho ^{+}}^{\alpha , \beta }\mathcal{F}(o,q)+\mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha , \beta } \mathcal{F}(o,\rho ) \bigr] \\ &\quad \supseteq \bigl[\mathcal{F}(o,\rho )+\mathcal{F}(o,q)+\mathcal{F}( \varsigma , \rho )+\mathcal{F}(\varsigma ,q) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h(\tau )h(\theta )+h(1-\tau )h(\theta )\\ &\qquad {}+h(\tau )h(1-\theta )+h(1- \tau )h(1-\theta ) \bigr]\,d\tau \,d\theta . \end{aligned} \end{aligned}$$
(3.5)

Using inequalities (3.3) and (3.5) completes the proof. □

Example 3.2

Let \(\triangle =[0,2]\times [0,2]\). Let \(h(\theta )=\theta \), \(\alpha =\beta =\frac{1}{2}\), and \(\mathcal{F}(\chi ,\gamma )= [(2-\sqrt{\chi })(2-\sqrt{\gamma }), (2+ \sqrt{\chi })(2+\sqrt{\gamma }) ]\). Then

$$\begin{aligned}& \frac{1}{\alpha \beta h^{2}(\frac{1}{2})}\mathcal{F} \biggl( \frac{o+\varsigma }{2}, \frac{\rho +q}{2} \biggr)=[16,144], \\& \begin{aligned} &\frac{\Gamma (\alpha )\Gamma (\beta )}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \bigl[\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta } \mathcal{F}( \varsigma ,q)+\mathfrak{J}_{o^{+},q^{-}}^{\alpha ,\beta }\mathcal{F}( \varsigma ,\rho )+\mathfrak{J}_{\varsigma ^{-},\rho ^{+}}^{\alpha , \beta }\mathcal{F}(o,q)+ \mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha , \beta }\mathcal{F}(o,\rho ) \bigr] \\ &\quad = \biggl[66-16\sqrt{2}-8\sqrt{2}\pi +2\pi +\frac{\pi ^{2}}{2}, 66+16 \sqrt{2}+8\sqrt{2}\pi +2\pi +\frac{\pi ^{2}}{2} \biggr], \end{aligned} \end{aligned}$$

and

$$ \begin{aligned} & \bigl[\mathcal{F}(o,\rho )+\mathcal{F}(o,q)+ \mathcal{F}(\varsigma , \rho )+\mathcal{F}(\varsigma ,q) \bigr] \\ &\qquad {}\times \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h(\tau )h(\theta )+h(1-\tau )h(\theta )\\ &\qquad {}+h(\tau )h(1-\theta )+h(1- \tau )h(1-\theta ) \bigr]\,d\tau \,d\theta \\ &\quad = [72-32\sqrt{2}, 72+32\sqrt{2} ]. \end{aligned} $$

Therefore

$$ \begin{aligned} {}[16,144]&\supseteq \biggl[66-16\sqrt{2}-8\sqrt{2}\pi +2\pi + \frac{\pi ^{2}}{2}, 66+16\sqrt{2}+8\sqrt{2}\pi +2\pi + \frac{\pi ^{2}}{2} \biggr]\\ &\supseteq [72-32 \sqrt{2}, 72+32\sqrt{2} ]. \end{aligned} $$

Consequently, Theorem 3.1 is verified.

Remark 3.3

If \(\underline{\mathcal{F}}=\overline{\mathcal{F}}\) and \(h(\theta )=\theta \), then we get Theorem 3 of [33]. If \(\underline{\mathcal{F}}=\overline{\mathcal{F}}\), \(h(\theta )=\theta \) and \(\alpha =\beta =1\), then we get Theorem 1 of [34].

Theorem 3.4

Let \(\mathcal{F}:\triangle \rightarrow \mathbb{R}_{\mathcal{I}}^{+}\) be such that \(\mathcal{F}=[\underline{\mathcal{F}},\overline{\mathcal{F}}]\) and \(\mathcal{F}\in \mathcal{ID}_{(\triangle )}\), and let \(h:[0,1]\rightarrow \mathbb{R}^{+}\). If \(\mathcal{F}\in SX(ch,\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\), then

$$\begin{aligned} \begin{aligned} &\frac{1}{h^{2} (\frac{1}{2} )} \mathcal{F} \biggl( \frac{o+\varsigma }{2},\frac{\rho +q}{2} \biggr) \\ &\quad \supseteq \frac{\Gamma (\alpha +1)}{2 h (\frac{1}{2} )(\varsigma -o)^{\alpha }} \biggl[\mathfrak{J}_{o^{+}}^{\alpha } \mathcal{F} \biggl(\varsigma , \frac{\rho +q}{2} \biggr)+\mathfrak{J}_{\varsigma^{-}}^{\alpha } \mathcal{F} \biggl(o,\frac{\rho +q}{2} \biggr) \biggr] \\ &\qquad {}+ \frac{\Gamma (\beta +1)}{2 h (\frac{1}{2} )(q-\rho )^{\beta }} \biggl[\mathfrak{J}_{\rho ^{+}}^{\beta } \mathcal{F} \biggl( \frac{o+\varsigma }{2},q \biggr)+\mathfrak{J}_{q^{-}}^{\beta } \mathcal{F} \biggl(\frac{o+\varsigma }{2},\rho \biggr) \biggr] \\ &\quad \supseteq \frac{ \Gamma (\alpha +1)\Gamma (\beta +1)}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }}\\ &\qquad {}\times \bigl[\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta } \mathcal{F}( \varsigma ,q)+\mathfrak{J}_{o^{+},q^{-}}^{\alpha ,\beta }\mathcal{F}( \varsigma ,\rho )+\mathfrak{J}_{\varsigma ^{-},\rho ^{+}}^{\alpha , \beta }\mathcal{F}(o,q)+ \mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha , \beta }F(o,\rho ) \bigr] \\ &\quad \supseteq \frac{\beta \Gamma (\alpha +1)}{(\varsigma -o)^{\alpha }} \bigl[\mathfrak{J}_{o^{+}}^{\alpha } \mathcal{F}(\varsigma ,\rho )+ \mathfrak{J}_{o^{+}}^{\alpha } \mathcal{F}(\varsigma ,q)+\mathfrak{J}_{ \varsigma ^{-}}^{\alpha }\mathcal{F}( \varsigma ,\rho )+\mathfrak{J}_{ \varsigma ^{-}}^{\alpha }\mathcal{F}(\varsigma ,q) \bigr]\\ &\qquad {}\times \int _{0}^{1} \theta ^{\beta -1} \bigl[h(\theta )+h(1-\theta ) \bigr]\,d\theta \\ &\qquad {}+\frac{\alpha \Gamma (\beta +1)}{(q-\rho )^{\beta }} \bigl[ \mathfrak{J}_{\rho ^{+}}^{\beta } \mathcal{F}(o,q)+\mathfrak{J}_{\rho^{+}}^{ \beta }\mathcal{F}(\varsigma ,q)+\mathfrak{J}_{q^{-}}^{\beta } \mathcal{F}(o,\rho )+ \mathfrak{J}_{q^{-}}^{\beta }\mathcal{F}( \varsigma ,\rho ) \bigr] \\ &\qquad {}\times\int _{0}^{1}\tau ^{\alpha -1} \bigl[h(\tau )+h(1- \tau ) \bigr]\,d\tau \\ &\quad \supseteq \alpha \beta \bigl[\mathcal{F}(o,\rho )+\mathcal{F}(o,q)+ \mathcal{F}(\varsigma ,\rho )+\mathcal{F}(\varsigma ,q) \bigr] \\ &\qquad {}\times\int _{0}^{1} \tau ^{\alpha -1} \bigl[h(\tau )+h(1-\tau ) \bigr]\,d\tau \int _{0}^{1} \theta ^{\beta -1} \bigl[h(\theta )+h(1-\theta ) \bigr]\,d\theta . \end{aligned} \end{aligned}$$
(3.6)

Proof

Using Theorem 2.7 and \(\mathcal{F}\in SX(ch,\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\), we get

$$\begin{aligned} \begin{aligned} \frac{1}{\beta h(\frac{1}{2})}\mathcal{F}_{\chi } \biggl( \frac{\rho +q}{2} \biggr)&\supseteq \frac{\Gamma (\beta )}{(q-\rho )^{\beta }} \bigl[\mathfrak{J}_{\rho ^{+}}^{ \beta } \mathcal{F}_{\chi }(q)+\mathfrak{J}_{q^{-}}^{\beta } \mathcal{F}_{ \chi }(\rho ) \bigr] \\ &\supseteq \bigl[\mathcal{F}_{\chi }(\rho )+\mathcal{F}_{\chi }(q) \bigr] \int _{0}^{1}\theta ^{\beta -1} \bigl[h(\theta )+h(1-\theta ) \bigr]\,d\theta , \end{aligned} \end{aligned}$$

that is,

$$\begin{aligned} \begin{aligned} &\frac{1}{\beta h(\frac{1}{2})} \mathcal{F} \biggl(\chi , \frac{\rho +q}{2} \biggr)\\ &\quad \supseteq \frac{1}{(q-\rho )^{\beta }} \biggl[ \int _{\rho }^{q}(q-\gamma )^{\beta -1}\mathcal{F}( \chi ,\gamma )\,d \gamma + \int _{\rho }^{q}(\gamma -\rho )^{\beta -1} \mathcal{F}(\chi , \gamma )\,d\gamma \biggr] \\ &\quad \supseteq \bigl[\mathcal{F}(\chi ,\rho )+\mathcal{F}(\chi ,q) \bigr] \int _{0}^{1}\theta ^{\beta -1} \bigl[h(\theta )+h(1-\theta ) \bigr]\,d\theta \end{aligned} \end{aligned}$$

for all \(\chi \in [o,\varsigma ]\). Moreover, we have

$$\begin{aligned} \begin{aligned} &\frac{1}{\beta (\varsigma -o)^{\alpha } h (\frac{1}{2} )} \int _{o}^{\varsigma }(\varsigma -\chi )^{\alpha -1} \mathcal{F} \biggl( \chi ,\frac{\rho +q}{2} \biggr)\,d\chi \\ &\quad \supseteq \frac{ 1}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \biggl[ \int _{o}^{\varsigma } \int _{\rho }^{q}(\varsigma -\chi )^{\alpha -1}(q- \gamma )^{\beta -1}\mathcal{F}(\chi ,\gamma )\,d\gamma \,d\chi \\ &\qquad {} + \int _{o}^{\varsigma } \int _{\rho }^{q}(\varsigma -\chi )^{ \alpha -1}( \gamma -\rho )^{\beta -1}\mathcal{F}(\chi ,\gamma )\,d \gamma \,d\chi \biggr] \\ &\quad \supseteq \frac{1}{(\varsigma -o)^{\alpha }} \int _{o}^{\varsigma } \int _{0}^{1}(\varsigma -\chi )^{\alpha -1} \bigl[\mathcal{F}(\chi , \rho )+\mathcal{F}(\chi ,q) \bigr]\theta ^{\beta -1} \bigl[h(\theta )+h(1- \theta ) \bigr]\,d\theta \,d\chi \end{aligned} \end{aligned}$$
(3.7)

and

$$\begin{aligned} \begin{aligned} &\frac{1}{\beta (\varsigma -o)^{\alpha } h (\frac{1}{2} )} \int _{o}^{\varsigma }(\chi -o)^{\alpha -1}\mathcal{F} \biggl(\chi , \frac{\rho +q}{2} \biggr)\,d\chi \\ &\quad \supseteq \frac{ 1}{(\varsigma -o)^{\alpha }(q-{\rho })^{\beta }} \biggl[ \int _{o}^{\varsigma } \int _{\rho }^{q}(\chi -o)^{\alpha -1}(q-\gamma )^{ \beta -1}\mathcal{F}(\chi ,\gamma )\,d\gamma \,d\chi \\ & \qquad {}+ \int _{o}^{\varsigma } \int _{\rho }^{q}( \chi -o)^{\alpha -1}(\gamma -\rho )^{\beta -1}\mathcal{F}(\chi , \gamma )\,d\gamma \,d\chi \biggr] \\ &\quad \supseteq \frac{1}{(\varsigma -o)^{\alpha }} \int _{o}^{\varsigma } \int _{0}^{1}(\chi -o)^{\alpha -1} \bigl[ \mathcal{F}(\chi ,\rho )+ \mathcal{F}(\chi ,q) \bigr]\theta ^{\beta -1} \bigl[h(\theta )+h(1- \theta ) \bigr]\,d\theta \,d\chi . \end{aligned} \end{aligned}$$
(3.8)

Similarly, we have

$$\begin{aligned} \begin{aligned} &\frac{1}{\alpha (q-\rho )^{\beta } h (\frac{1}{2} )} \int _{ \rho }^{q}(q-\gamma )^{\beta -1}\mathcal{F} \biggl( \frac{o+\varsigma }{2},\gamma \biggr)\,d\gamma \\ &\quad \supseteq \frac{ 1}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \biggl[ \int _{o}^{\varsigma } \int _{\rho }^{q}(\varsigma -\chi )^{\alpha -1}(q- \gamma )^{\beta -1}\mathcal{F}(\chi ,\gamma )\,d\gamma \,d\chi \\ &\qquad {} + \int _{o}^{\varsigma } \int _{\rho }^{q}(\chi -o)^{\alpha -1}(q- \gamma )^{\beta -1}\mathcal{F}(\chi ,\gamma )\,d\gamma \,d\chi \biggr] \\ &\quad \supseteq \frac{1}{(q-\rho )^{\beta }} \int _{\rho }^{q} \int _{0}^{1}(q- \gamma )^{\beta -1} \bigl[ \mathcal{F}(o,\gamma )+\mathcal{F}( \varsigma ,\gamma ) \bigr]\tau ^{\alpha -1} \bigl[h(\tau )+h(1- \tau ) \bigr]\,d\tau \,d\gamma \end{aligned} \end{aligned}$$
(3.9)

and

$$\begin{aligned} \begin{aligned} &\frac{1}{\alpha (q-\rho )^{\beta } h (\frac{1}{2} )} \int _{ \rho }^{q}(\gamma -\rho )^{\beta -1} \mathcal{F} \biggl( \frac{o+\varsigma }{2},\gamma \biggr)\,d\gamma \\ &\quad \supseteq \frac{ 1}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \biggl[ \int _{o}^{\varsigma } \int _{\rho }^{q}(\varsigma -\chi )^{\alpha -1}( \gamma -\rho )^{\beta -1}\mathcal{F}(\chi ,\gamma )\,d\gamma \,d\chi \\ &\qquad {} + \int _{o}^{\varsigma } \int _{\rho }^{q}(\chi -o)^{\alpha -1}( \gamma -\rho )^{\beta -1}\mathcal{F}(\chi ,\gamma )\,d\gamma \,d\chi \biggr] \\ &\quad \supseteq \frac{1}{(q-\rho )^{\beta }} \int _{\rho }^{q} \int _{0}^{1}( \gamma -\rho )^{\beta -1} \bigl[ \mathcal{F}(o,\gamma )+\mathcal{F}( \varsigma ,\gamma ) \bigr]\tau ^{\alpha -1} \bigl[h(\tau )+h(1- \tau ) \bigr]\,d\tau \,d\gamma . \end{aligned} \end{aligned}$$
(3.10)

Summing inequalities (3.7)–(3.10), we have

$$\begin{aligned} \begin{aligned} &\frac{\Gamma (\alpha +1)}{2 h (\frac{1}{2} )(\varsigma -o)^{\alpha }} \biggl[ \mathfrak{J}_{o^{+}}^{\alpha }\mathcal{F} \biggl(\varsigma , \frac{\rho +q}{2} \biggr)+\mathfrak{J}_{\varsigma ^{-}}^{\alpha } \mathcal{F} \biggl(o,\frac{\rho +q}{2} \biggr) \biggr] \\ &\qquad {}+ \frac{\Gamma (\beta +1)}{2 h (\frac{1}{2} )(q-\rho )^{\beta }} \biggl[\mathfrak{J}_{\rho ^{+}}^{\beta } \mathcal{F} \biggl( \frac{o+\varsigma }{2},q \biggr)+\mathfrak{J}_{q^{-}}^{\beta } \mathcal{F} \biggl(\frac{o+\varsigma }{2},\varsigma \biggr) \biggr] \\ &\quad \supseteq \frac{ \Gamma (\alpha +1)\Gamma (\beta +1)}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \bigl[\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta } \mathcal{F}( \varsigma ,q)+\mathfrak{J}_{o^{+},q^{-}}^{\alpha ,\beta }\mathcal{F}( \varsigma ,\rho )+\mathfrak{J}_{\varsigma ^{-},\rho ^{+}}^{\alpha , \beta }\mathcal{F}(o,q)+ \mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha , \beta }\mathcal{F}(o,\rho ) \bigr] \\ &\quad \supseteq \frac{\beta \Gamma (\alpha +1)}{(\varsigma -o)^{\alpha }} \bigl[\mathfrak{J}_{o^{+}}^{\alpha } \mathcal{F}(\varsigma ,\rho )+ \mathfrak{J}_{o^{+}}^{\alpha } \mathcal{F}(\varsigma ,q)+\mathfrak{J}_{ \varsigma ^{-}}^{\alpha }\mathcal{F}( \varsigma ,\rho )+\mathfrak{J}_{ \varsigma ^{-}}^{\alpha }\mathcal{F}(\varsigma ,q) \bigr]\\ &\qquad {}\times \int _{0}^{1} \theta ^{\beta -1} \bigl[h(\theta )+h(1-\theta ) \bigr]\,d\theta \\ &\qquad {}+\frac{\alpha \Gamma (\beta +1)}{(q-\rho )^{\beta }} \bigl[ \mathfrak{J}_{\rho ^{+}}^{\beta } \mathcal{F}(o,q)+\mathfrak{J}_{\rho^{+}}^{ \beta }\mathcal{F}(\varsigma ,q)+\mathfrak{J}_{q^{-}}^{\beta } \mathcal{F}(o,\varsigma )+ \mathfrak{J}_{q^{-}}^{\beta }\mathcal{F}( \varsigma ,\rho ) \bigr]\\ &\qquad {}\times \int _{0}^{1}\tau ^{\alpha -1} \bigl[h(\tau )+h(1- \tau ) \bigr]\,d\tau , \end{aligned} \end{aligned}$$

which gives the second and third inequalities in (3.6).

Using the first inequality in (2.1), we get

$$\begin{aligned} \begin{aligned} \frac{1}{ h^{2} (\frac{1}{2} )} \mathcal{F} \biggl( \frac{o+\varsigma }{2},\frac{\rho +q}{2} \biggr)\supseteq \frac{\Gamma (\alpha +1)}{h (\frac{1}{2} )(\varsigma -o)^{\alpha }} \biggl[\mathfrak{J}_{\varsigma ^{-}}^{\alpha }\mathcal{F} \biggl(o, \frac{\rho +q}{2} \biggr)+\mathfrak{J}_{o^{+}}^{\alpha } \mathcal{F} \biggl(\varsigma ,\frac{\rho +q}{2} \biggr) \biggr] \end{aligned} \end{aligned}$$
(3.11)

and

$$\begin{aligned} \begin{aligned} \frac{1}{ h^{2} (\frac{1}{2} )} \mathcal{F} \biggl( \frac{o+\varsigma }{2},\frac{\rho +q}{2} \biggr)\supseteq \frac{\Gamma (\beta +1)}{h (\frac{1}{2} )(q-\rho )^{\beta }} \biggl[\mathfrak{J}_{q^{-}}^{\beta }\mathcal{F} \biggl( \frac{o+\varsigma }{2},\rho \biggr)+\mathfrak{J}_{\rho ^{+}}^{\beta } \mathcal{F} \biggl(\frac{o+\varsigma }{2},q \biggr) \biggr]. \end{aligned} \end{aligned}$$
(3.12)

Summing inequalities (3.11) and (3.12), we get the first inequality in (3.6).

Using the second inequality in (2.1), we also state

$$\begin{aligned}& \frac{\Gamma (\alpha )}{(\varsigma -o)^{\alpha }} \bigl[\mathfrak{J}_{ \varsigma ^{-}}^{\alpha }\mathcal{F} (o, \rho )+ \mathfrak{J}_{o^{+}}^{\alpha }\mathcal{F} (\varsigma ,\rho ) \bigr]\supseteq \bigl[\mathcal{F}(o,\rho )+\mathcal{F}( \varsigma ,\rho ) \bigr] \int _{0}^{1}\tau ^{\alpha -1} \bigl[h(\tau )+h(1- \tau ) \bigr]\,d\tau , \\& \frac{\Gamma (\alpha )}{(\varsigma -o)^{\alpha }} \bigl[\mathfrak{J}_{ \varsigma ^{-}}^{\alpha }\mathcal{F} (o,q )+\mathfrak{J}_{o^{+}}^{ \alpha }\mathcal{F} (\varsigma ,q ) \bigr]\supseteq \bigl[ \mathcal{F}(o,q)+\mathcal{F}(\varsigma ,q) \bigr] \int _{0}^{1}\tau ^{ \alpha -1} \bigl[h(\tau )+h(1-\tau ) \bigr]\,d\tau , \\& \frac{\Gamma (\beta )}{(q-\rho )^{\beta }} \bigl[\mathfrak{J}_{q^{-}}^{ \beta }\mathcal{F} (o,\rho )+\mathfrak{J}_{\rho ^{+}}^{ \beta }\mathcal{F} (o,q ) \bigr] \supseteq \bigl[ \mathcal{F}o,\rho )+\mathcal{F}(o,q) \bigr] \int _{0}^{1}\theta ^{ \beta -1} \bigl[h(\theta )+h(1-\theta ) \bigr]\,d\theta , \end{aligned}$$

and

$$\begin{aligned} \frac{\Gamma (\beta )}{(q-\rho )^{\beta }} \bigl[\mathfrak{J}_{q^{-}}^{ \beta } \mathcal{F} (\varsigma ,\rho )+\mathfrak{J}_{\rho ^{+}}^{ \beta }F ( \varsigma ,q ) \bigr]\supseteq \bigl[F( \varsigma ,\rho )+F(\varsigma ,q) \bigr] \int _{0}^{1}\theta ^{\beta -1} \bigl[h(\theta )+h(1-\theta ) \bigr]\,d\theta , \end{aligned}$$

which gives the last inequality in (3.6). This completes the proof. □

Remark 3.5

If \(\underline{\mathcal{F}}=\overline{\mathcal{F}}\) and \(h(\theta )=\theta \), then we get Theorem 4 of [33]. If \(\alpha =\beta =1\), then we get Theorem 3.5 of [25]. If \(\alpha =\beta =1\) and \(h(\theta )=\theta \), then we get Theorem 7 of [26].

Theorem 3.6

Let \(\mathcal{F},\mathcal{G}:\triangle \rightarrow \mathbb{R}_{ \mathcal{I}}^{+}\) be such that \(\mathcal{F}=[\underline{\mathcal{F}},\overline{\mathcal{F}}]\), \(\mathcal{G}=[\underline{\mathcal{G}},\overline{\mathcal{G}}]\), and \(\mathcal{F}\mathcal{G}\in \mathcal{ID}_{(\triangle )}\), and let \(h:[0,1]\rightarrow \mathbb{R}^{+}\). If \(\mathcal{F}\in SX(ch_{1},\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\) and \(\mathcal{G}\in SX(ch_{2},\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\), then

$$\begin{aligned} \begin{aligned} &\frac{\Gamma (\alpha )\Gamma (\beta )}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \bigl[ \mathfrak{J}^{\alpha ,\beta }_{\varsigma ^{-},\rho ^{+}} \mathcal{F}(o,q)\mathcal{G}(o,q)+ { \mathfrak{J}}^{\alpha ,\beta }_{ \varsigma ^{-},q^{-}}\mathcal{F}(o,\rho )\mathcal{G}(o,\rho ) \\ & \qquad {}+\mathfrak{J}^{\alpha ,\beta }_{o^{+}, \rho ^{+}}\mathcal{F}( \varsigma ,q)\mathcal{G}(\varsigma ,q)+ { \mathfrak{J}}^{\alpha ,\beta }_{o^{+},q^{-}} \mathcal{F}(\varsigma , \rho )\mathcal{G}(\varsigma ,\rho ) \bigr] \\ &\quad \supseteq \mathcal{M}(o,\varsigma ,\rho ,q) \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h_{1}(1-\tau )h_{1}(1- \theta )h_{2}(1-\tau )h_{2}(1-\theta ) \\ &\qquad {}+h_{1}(1-\tau )h_{1}(\theta ) h_{2}(1-\tau )h_{2}(\theta )+h_{1}(\tau )h_{1}(1- \theta )h_{2}(\tau )h_{2}(1-\theta )\\ &\qquad {}+h_{1}(\tau )h_{1}(\theta )h_{2}( \tau )h_{2}(\theta ) \bigr]\,d\tau \,d\theta \\ &\qquad{} + \mathcal{N}(o,\varsigma ,\rho ,q) \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h_{1}(1-\tau )h_{1}(\theta )h_{2}(1- \tau )h_{2}(1-\theta ) \\ &\qquad {}+h_{1}(1-\tau )h_{1}(1-\theta ) h_{2}(1-\tau )h_{2}(\theta )+h_{1}(\tau )h_{1}(1- \theta )h_{2}(\tau )h_{2}(\theta )\\ &\qquad {}+h_{1}(\tau )h_{1}(\theta )h_{2}( \tau )h_{2}(1-\theta ) \bigr]\,d\tau \,d\theta \\ &\qquad{} + \mathcal{P}(o,\varsigma ,\rho ,q) \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(1-\theta )h_{2}(1- \tau )h_{2}(1-\theta ) \\ &\qquad {}+h_{1}(1-\tau )h_{1}(1-\theta ) h_{2}(\tau )h_{2}(1-\theta )+h_{1}(\tau )h_{1}(\theta )h_{2}(1- \tau )h_{2}(\theta )\\ &\qquad {}+h_{1}(1-\tau )h_{1}(\theta )h_{2}(\tau )h_{2}( \theta ) \bigr]\,d\tau \,d\theta \\ &\qquad{} + \mathcal{Q}(o,\varsigma ,\rho ,q) \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta )h_{2}(1- \tau )h_{2}(1-\theta ) \\ &\qquad {}+h_{1}(\tau )h_{1}(1-\theta ) h_{2}(1-\tau )h_{2}(\theta )+h_{1}(1-\tau )h_{1}( \theta )h_{2}(\tau )h_{2}(1-\theta )\\ &\qquad {}+h_{1}(\tau )h_{1}(\theta )h_{2}(1- \tau )h_{2}(1-\theta ) \bigr]\,d\tau \,d\theta , \end{aligned} \end{aligned}$$
(3.13)

where

$$\begin{aligned}& \mathcal{M}(o,\varsigma ,\rho ,q)=\mathcal{F}(o,\rho )\mathcal{G}(o, \rho )+ \mathcal{F}(\varsigma ,\rho )\mathcal{G}(\varsigma ,\rho )+ \mathcal{F}(o,q) \mathcal{G}(o,q)+\mathcal{F}(\varsigma ,q)\mathcal{G}( \varsigma ,q), \\& \mathcal{N}(o,\varsigma ,\rho ,q)=\mathcal{F}(o,\rho )\mathcal{G}(o,q)+ \mathcal{F}(\varsigma ,\rho )\mathcal{G}(\varsigma ,q)+\mathcal{F}(o,q) \mathcal{G}(o,\rho )+\mathcal{F}(\varsigma ,q)\mathcal{G}(\varsigma , \rho ), \\& \mathcal{P}(o,\varsigma ,\rho ,q)=\mathcal{F}(o,\rho )\mathcal{G}(\varsigma, \rho )+ \mathcal{F}(\varsigma ,\rho )\mathcal{G}(o,\rho )+\mathcal{F}(o,q) \mathcal{G}( \varsigma ,q)+\mathcal{F}(\varsigma ,q)\mathcal{G}(o,q), \\& \mathcal{Q}(o,\varsigma ,\rho ,q)=\mathcal{F}(o,\rho )\mathcal{G}( \varsigma ,q)+\mathcal{F}(\varsigma ,\rho )\mathcal{G}(o,q)+ \mathcal{F}(o,q)\mathcal{G}( \varsigma ,\rho )+\mathcal{F}(\varsigma ,q) \mathcal{G}(o,\rho ). \end{aligned}$$

Proof

Since \(\mathcal{F}\in SX(ch_{1},\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\) and \(\mathcal{G}\in SX(ch_{2},\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\), we have

$$\begin{aligned}& \begin{aligned} &\mathcal{F}\bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1-\theta ) q\bigr)\\ &\quad \supseteq h_{1}(\tau )h_{1}( \theta )\mathcal{F}(o,\rho )+h_{1}(\tau )h_{1}(1- \theta ) \mathcal{F}(o,q) \\ & \qquad {}+h_{1}(1-\tau )h_{1}(\theta )\mathcal{F}( \varsigma ,\rho )+h_{1}(1- \tau )h_{1}(1-\theta ) \mathcal{F}(\varsigma ,q), \end{aligned} \\& \begin{aligned} &\mathcal{F}\bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho + \theta q\bigr)\\ &\quad \supseteq h_{1}(\tau )h_{1}(1-\theta ) \mathcal{F}(o,\rho )+h_{1}( \tau )h_{1}(\theta ) \mathcal{F}(o,q) \\ & \qquad {}+h_{1}(1-\tau )h_{1}(1-\theta )\mathcal{F}( \varsigma ,\rho )+h_{1}(1- \tau )h_{1}(\theta )\mathcal{F}( \varsigma ,q), \end{aligned} \\& \begin{aligned} &\mathcal{F}\bigl((1-\tau ) o+\tau \varsigma ,\theta \rho +(1- \theta ) q\bigr)\\ &\quad \supseteq h_{1}(1-\tau )h_{1}(\theta ) \mathcal{F}(o,\rho )+h_{1}(1- \tau )h_{1}(1-\theta ) \mathcal{F}(o,q) \\ & \qquad {}+h_{1}(\tau )h_{1}(\theta )\mathcal{F}(\varsigma ,\rho )+h_{1}( \tau )h_{1}(1-\theta )\mathcal{F}(\varsigma ,q), \end{aligned} \\& \begin{aligned} &\mathcal{F}\bigl((1-\tau )o+\tau \varsigma ,(1-\theta )\rho + \theta q\bigr)\\ &\quad \supseteq h_{1}(1-\tau )h_{1}(1-\theta ) \mathcal{F}(o,\rho )+h_{1}(1- \tau )h_{1}(\theta ) \mathcal{F}(o,q) \\ & \qquad {}+h_{1}(\tau )h_{1}(1-\theta )\mathcal{F}( \varsigma ,\rho )+h_{1}( \tau )h_{1}(\theta )\mathcal{F}( \varsigma ,q), \end{aligned} \end{aligned}$$

and

$$\begin{aligned}& \begin{aligned}& \mathcal{G}\bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1-\theta ) q\bigr)\\ &\quad \supseteq h_{2}(\tau )h_{2}( \theta )\mathcal{G}(o,\rho )+h_{2}(\tau )h_{2}(1- \theta ) \mathcal{G}(o,q) \\ & \qquad {}+h_{2}(1-\tau )h_{2}(\theta )\mathcal{G}( \varsigma ,\rho )+h_{2}(1- \tau )h_{2}(1-\theta ) \mathcal{G}(\varsigma ,q), \end{aligned} \\& \begin{aligned} &\mathcal{G}\bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho + \theta q\bigr)\\ &\quad \supseteq h_{2}(\tau )h_{2}(1-\theta ) \mathcal{G}(o,\rho )+h_{2}( \tau )h_{2}(\theta ) \mathcal{G}(o,q) \\ & \qquad {}+h_{2}(1-\tau )h_{2}(1-\theta )\mathcal{G}( \varsigma ,\rho )+h_{2}(1- \tau )h_{2}(\theta )\mathcal{G}( \varsigma ,q), \end{aligned} \\& \begin{aligned} &\mathcal{G}\bigl((1-\tau ) o+\tau \varsigma ,\theta \rho +(1- \theta ) q\bigr)\\ &\quad \supseteq h_{2}(1-\tau )h_{2}(\theta ) \mathcal{G}(o,\rho )+h_{2}(1- \tau )h_{2}(1-\theta ) \mathcal{G}(o,q) \\ & \qquad {}+h_{2}(\tau )h_{2}(\theta )\mathcal{G}(\varsigma ,\rho )+h_{2}( \tau )h_{2}(1-\theta )\mathcal{G}(\varsigma ,q), \end{aligned} \\& \begin{aligned} &\mathcal{G}\bigl((1-\tau )o+\tau \varsigma ,(1-\theta )\rho + \theta q\bigr)\\ &\quad \supseteq h_{2}(1-\tau )h_{2}(1-\theta ) \mathcal{G}(o,\rho )+h_{2}(1- \tau )h_{2}(\theta ) \mathcal{G}(o,q) \\ & \qquad {}+h_{2}(\tau )h_{2}(1-\theta )\mathcal{G}( \varsigma ,\rho )+h_{2}( \tau )h_{2}(\theta )\mathcal{G}( \varsigma ,q). \end{aligned} \end{aligned}$$

Since \(\mathcal{F},\mathcal{G}\in \mathbb{R}_{\mathcal{I}}^{+}\), we have

$$\begin{aligned} &\mathcal{F}\bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1-\theta ) q\bigr) \mathcal{G}\bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1-\theta ) q\bigr) \\ &\qquad {}+\mathcal{F}\bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho +\theta q \bigr) \mathcal{G}\bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho +\theta q\bigr) \\ &\qquad {}+\mathcal{F}\bigl((1-\tau ) o+\tau \varsigma ,\theta \rho +(1-\theta ) q\bigr) \mathcal{G}\bigl((1-\tau ) o+\tau \varsigma ,\theta \rho +(1-\theta ) q \bigr) \\ &\qquad {}+\mathcal{F}\bigl((1-\tau )o+\tau \varsigma ,(1-\theta )\rho +\theta q \bigr) \mathcal{G}\bigl((1-\tau )o+\tau \varsigma ,(1-\theta )\rho +\theta q\bigr) \\ &\quad \supseteq \mathcal{M}(o,\varsigma ,\rho ,q) \\ &\qquad {}\times \bigl[h_{1}(1-\tau )h_{1}(1- \theta )h_{2}(1-\tau )h_{2}(1-\theta )+h_{1}(1-\tau )h_{1}(\theta )h_{2}(1- \tau )h_{2}(\theta ) \\ &\qquad {}+h_{1}(\tau )h_{1}(1-\theta )h_{2}(\tau )h_{2}(1-\theta )+h_{1}( \tau )h_{1}(\theta )h_{2}(\tau )h_{2}(\theta ) \bigr] \\ &\qquad + \mathcal{N}(o,\varsigma ,\rho ,q) \bigl[h_{1}(1-\tau )h_{1}( \theta )h_{2}(1-\tau )h_{2}(1-\theta ){+}h_{1}(1-\tau )h_{1}(1-\theta )h_{2}(1- \tau )h_{2}(\theta ) \\ &\qquad {}+h_{1}(\tau )h_{1}(1-\theta )h_{2}(\tau )h_{2}(\theta )+h_{1}( \tau )h_{1}(\theta )h_{2}(\tau )h_{2}(1-\theta ) \bigr] \\ &\qquad{} + \mathcal{P}(o,\varsigma ,\rho ,q) \bigl[h_{1}(\tau )h_{1}(1- \theta )h_{2}(1-\tau )h_{2}(1-\theta ){+}h_{1}(1-\tau )h_{1}(1-\theta )h_{2}( \tau )h_{2}(1-\theta ) \\ &\qquad {}+h_{1}(\tau )h_{1}(\theta )h_{2}(1-\tau )h_{2}(\theta )+h_{1}(1- \tau )h_{1}(\theta )h_{2}(\tau )h_{2}(\theta ) \bigr] \\ &\qquad{} + \mathcal{Q}(o,\varsigma ,\rho ,q) \bigl[h_{1}(\tau )h_{1}( \theta )h_{2}(1-\tau )h_{2}(1-\theta )+h_{1}(\tau )h_{1}(1-\theta )h_{2}(1- \tau )h_{2}(\theta ) \\ &\qquad {}+h_{1}(1-\tau )h_{1}(\theta )h_{2}(\tau )h_{2}(1-\theta )+h_{1}( \tau )h_{1}(\theta )h_{2}(1-\tau )h_{2}(1-\theta ) \bigr]. \end{aligned}$$

Moreover, we have

$$\begin{aligned}& \begin{gathered} \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \mathcal{F}\bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1- \theta ) q\bigr)\\ \qquad {}\times \mathcal{G}\bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1- \theta ) q\bigr)\,d \tau \,d\theta \\ \qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \mathcal{F}\bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho + \theta q\bigr)\\ \qquad {}\times \mathcal{G}\bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho +\theta q\bigr)\,d \tau \,d\theta \\ \qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \mathcal{F}\bigl((1-\tau )o+\tau \varsigma ,\theta \rho +(1- \theta ) q\bigr)\\ \qquad {}\times \mathcal{G}\bigl((1-\tau )o+\tau \varsigma ,\theta \rho +(1- \theta ) q\bigr)\,d \tau \,d\theta \\ \qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \mathcal{F}\bigl((1-\tau )o+\tau \varsigma ,(1-\theta )\rho + \theta q\bigr)\\ \qquad {}\times \mathcal{G}\bigl((1-\tau )o+\tau \varsigma ,(1-\theta )\rho +\theta q\bigr)\,d \tau \,d\theta \\ \quad \supseteq \mathcal{M}(o,\varsigma ,\rho ,q) \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h_{1}(1-\tau )h_{1}(1- \theta )h_{2}(1-\tau )h_{2}(1-\theta ) \\ \qquad {}+h_{1}(1-\tau )h_{1}(\theta ) h_{2}(1-\tau )h_{2}(\theta )+h_{1}(\tau )h_{1}(1- \theta )h_{2}(\tau )h_{2}(1-\theta )\\ \qquad {}+h_{1}(\tau )h_{1}(\theta )h_{2}( \tau )h_{2}(\theta ) \bigr]\,d\tau \,d\theta \\ \qquad{} + \mathcal{N}(o,\varsigma ,\rho ,q) \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h_{1}(1-\tau )h_{1}(\theta )h_{2}(1- \tau )h_{2}(1-\theta ) \\ \qquad {}+h_{1}(1-\tau )h_{1}(1-\theta ) h_{2}(1-\tau )h_{2}(\theta )+h_{1}(\tau )h_{1}(1- \theta )h_{2}(\tau )h_{2}(\theta )\\ \qquad {}+h_{1}(\tau )h_{1}(\theta )h_{2}( \tau )h_{2}(1-\theta ) \bigr]\,d\tau \,d\theta \\ \qquad{} + \mathcal{P}(o,\varsigma ,\rho ,q) \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(1-\theta )h_{2}(1- \tau )h_{2}(1-\theta ) \\ \qquad {}+h_{1}(1-\tau )h_{1}(1-\theta ) h_{2}(\tau )h_{2}(1-\theta )+h_{1}(\tau )h_{1}(\theta )h_{2}(1- \tau )h_{2}(\theta )\\ \qquad {}+h_{1}(1-\tau )h_{1}(\theta )h_{2}(\tau )h_{2}( \theta ) \bigr]\,d\tau \,d\theta \\ \qquad{} + \mathcal{Q}(o,\varsigma ,\rho ,q) \int _{0}^{1} \int _{0}^{1} \tau ^{\alpha -1}\theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta )h_{2}(1- \tau )h_{2}(1-\theta ) \\ \qquad {}+h_{1}(\tau )h_{1}(1-\theta ) h_{2}(1-\tau )h_{2}(\theta )+h_{1}(1-\tau )h_{1}( \theta )h_{2}(\tau )h_{2}(1-\theta )\\ \qquad {}+h_{1}(\tau )h_{1}(\theta )h_{2}(1- \tau )h_{2}(1-\theta ) \bigr]\,d\tau \,d\theta . \end{gathered} \end{aligned}$$
(3.14)

By Definition 2.8 we get

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \mathcal{F}\bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1- \theta ) q\bigr) \\& \qquad {}\times \mathcal{G}\bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1- \theta ) q\bigr)\,d \tau \,d\theta \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \mathcal{F}\bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho + \theta q\bigr) \\& \qquad {}\times \mathcal{G}\bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho +\theta q\bigr)\,d \tau \,d\theta \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \mathcal{F}\bigl((1-\tau )o+\tau \varsigma ,\theta \rho +(1- \theta ) q\bigr) \\ & \\ & \qquad {}\times \mathcal{G}\bigl((1-\tau )o+\tau \varsigma ,\theta \rho +(1- \theta ) q\bigr)\,d \tau \,d\theta \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1}\tau ^{\alpha -1}\theta ^{\beta -1} \mathcal{F}\bigl((1-\tau )o+\tau \varsigma ,(1-\theta )\rho + \theta q\bigr) \\& \qquad {}\times \mathcal{G}\bigl((1-\tau )o+\tau \varsigma ,(1-\theta )\rho +\theta q\bigr)\,d \tau \,d\theta \\& \quad =\frac{\Gamma (\alpha )\Gamma (\beta )}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \bigl[\mathfrak{J}^{\alpha ,\beta }_{\varsigma ^{-},\rho^{+}} \mathcal{F}(o,q) \mathcal{G}(o,q)+ {\mathfrak{J}}^{\alpha ,\beta }_{\varsigma ^{-},q^{-}} \mathcal{F}(o,\rho )\mathcal{G}(o,\rho ) \\& \qquad {}+\mathfrak{J}^{\alpha ,\beta }_{o^{+},\rho^{+}} \mathcal{F}( \varsigma ,q)\mathcal{G}(\varsigma ,q)+ {\mathfrak{J}}^{ \alpha ,\beta }_{o^{+},q^{-}} \mathcal{F}(\varsigma ,\rho )\mathcal{G}( \varsigma ,\rho ) \bigr]. \end{aligned}$$
(3.15)

From inequalities (3.14)–(3.15) we obtain inequalities (3.13). □

Remark 3.7

If \(\alpha =\beta =1\) and \(h(\theta )=\theta \), then we get Theorem 8 of [26]. If \(\underline{\mathcal{F}}=\overline{\mathcal{F}}\), \(h(\theta )=\theta \), and \(\alpha =\beta =1\), then we get Theorem 4 of [35].

Theorem 3.8

Let \(\mathcal{F},\mathcal{G}:[o,\varsigma ]\times [\rho ,q]\rightarrow \mathbb{R}_{\mathcal{I}}^{+}\) be such that \(\mathcal{F}=[\underline{\mathcal{F}},\overline{\mathcal{F}}]\), \(\mathcal{G}=[\underline{\mathcal{G}},\overline{\mathcal{G}}]\), and \(\mathcal{F}\mathcal{G}\in \mathcal{ID}_{(\triangle )}\), and let \(h:[0,1]\rightarrow \mathbb{R}^{+}\). If \(\mathcal{F}\in SX(ch_{1},\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\) and \(\mathcal{G}\in SX(ch_{2},\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\), then

$$\begin{aligned}& \frac{1}{2\alpha \beta {h_{1}}^{2}(\frac{1}{2}){h_{2}}^{2}(\frac{1}{2})} \mathcal{F} \biggl(\frac{o+\varsigma }{2},\frac{\rho +q}{2} \biggr) \mathcal{G} \biggl( \frac{o+\varsigma }{2},\frac{\rho +q}{2} \biggr) \\& \quad \supseteq \frac{\Gamma (\alpha )\Gamma (\beta )}{2(\varsigma -o)^{\alpha }(q-\rho )^{\beta }} \bigl[\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta } \mathcal{F}( \varsigma ,q)\mathcal{G}(\varsigma ,q)+\mathfrak{J}_{o^{+},q^{-}}^{ \alpha ,\beta } \mathcal{F}(\varsigma ,\rho )\mathcal{G}(\varsigma , \rho ) \\& \qquad {}+\mathfrak{J}_{\varsigma ^{-},\rho ^{+}}^{ \alpha ,\beta }\mathcal{F}(o,q) \mathcal{G}(o,q)+\mathfrak{J}_{ \varsigma ^{-},q^{-}}^{\alpha ,\beta }\mathcal{F}(o,\rho ) \mathcal{G}(o, \rho ) \bigr] \\& \qquad {}+\mathcal{M}(o,\varsigma ,\rho ,q) \int _{0}^{1}\tau ^{\alpha -1}\,d \tau \int _{0}^{1}\theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta ) \bigl[h_{2}(\tau )h_{2}(1-\theta ) \\& \qquad {}+h_{2}(1-\tau )h_{2}(\theta )+h_{2}(1- \tau )h_{2}(1-\theta ) \bigr] \\& \qquad {}+h_{1}(\tau )h_{1}(1-\theta ) \bigl[h_{2}(\tau )h_{2}(\theta )+h_{2}(1- \tau )h_{2}(1-\theta )+h_{2}(1-\tau )h_{2}(\theta ) \bigr] \bigr]\,d \theta \\& \qquad {}+\mathcal{N}(o,\varsigma ,\rho ,q) \int _{0}^{1}\tau ^{\alpha -1}\,d \tau \int _{0}^{1}\theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta ) \bigl[h_{2}(\tau )h_{2}(\theta ) \\& \qquad {}+h_{2}(1-\tau )h_{2}(1-\theta )+h_{2}(1- \tau )h_{2}(\theta ) \bigr] \\& \qquad {}+h_{1}(\tau )h_{1}(1-\theta ) \bigl[h_{2}(\tau )h_{2}(1- \theta )+h_{2}(1-\tau )h_{2}(\theta )+h_{2}(1-\tau )h_{2}(1-\theta ) \bigr] \bigr]\,d\theta \\& \qquad {}+\mathcal{P}(o,\varsigma ,\rho ,q) \int _{0}^{1}\tau ^{\alpha -1}\,d \tau \int _{0}^{1}\theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta ) \bigl[h_{2}(1- \tau )h_{2}(1-\theta ) \\& \qquad {}+h_{2}(\tau )h_{2}(\theta )+h_{2}( \tau )h_{2}(1-\theta ) \bigr] \\& \qquad {}+h_{1}(\tau )h_{1}(1-\theta ) \bigl[h_{2}(1-\tau )h_{2}( \theta )+h_{2}(\tau )h_{2}(1-\theta )+h_{2}(\tau )h_{2}(\theta ) \bigr] \bigr]\,d\theta \\& \qquad {}+\mathcal{Q}(o,\varsigma ,\rho ,q) \int _{0}^{1}\tau ^{\alpha -1}\,d \tau \int _{0}^{1}\theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta ) \bigl[h_{2}(1- \tau )h_{2}(\theta ) \\& \qquad +h_{2}(\tau )h_{2}(1-\theta )+h_{2}( \tau )h_{2}(\theta ) \bigr] \\& \qquad {}+h_{1}(\tau )h_{1}(1-\theta ) \bigl[h_{2}(1-\tau )h_{2}(1- \theta )+h_{2}(\tau )h_{2}(\theta )+h_{2}(\tau )h_{2}(1-\theta ) \bigr] \bigr]\,d\theta . \end{aligned}$$
(3.16)

Proof

Since \(\mathcal{F}\in SX(ch_{1},\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\) and \(\mathcal{G}\in SX(ch_{2},\triangle ,\mathbb{R}_{\mathcal{I}}^{+})\), we have

$$\begin{aligned}& \mathcal{F} \biggl( \frac{o+\varsigma }{2},\frac{\rho +q}{2} \biggr) \mathcal{G} \biggl(\frac{o+\varsigma }{2}, \frac{\rho +q}{2} \biggr) \\& \quad =\mathcal{F} \biggl(\frac{\tau o+(1-\tau )\varsigma }{2}+ \frac{(1-\tau )o+\tau \varsigma }{2}, \frac{\theta \rho +(1-\theta )q}{2}+ \frac{(1-\theta )\rho +\theta q}{2} \biggr) \\& \qquad {}\times \mathcal{G} \biggl(\frac{\tau o+(1-\tau )\varsigma }{2}+ \frac{(1-\tau )o+\tau \varsigma }{2}, \frac{\theta \rho +(1-\theta )q}{2}+ \frac{(1-\theta )\rho +\theta q}{2} \biggr) \\& \quad \supseteq {h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr)\\& \qquad {}\times \bigl[ \mathcal{F} \bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1-\theta )q \bigr)+ \mathcal{F} \bigl((1-\tau )o+\tau \varsigma ,\theta \rho +(1- \theta )q \bigr) \\& \qquad {}+\mathcal{F} \bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho + \theta q \bigr)+\mathcal{F} \bigl((1-\tau )o+\tau \varsigma ,(1-\theta ) \rho + \theta q \bigr) \bigr] \\& \qquad {}\times \bigl[\mathcal{G} \bigl(\tau o+(1-\tau )\varsigma , \theta \rho +(1-\theta )q \bigr)+\mathcal{G} \bigl((1-\tau )o+\tau \varsigma , \theta \rho +(1-\theta )q \bigr) \\& \qquad {}+\mathcal{G} \bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho + \theta q \bigr)+\mathcal{G} \bigl((1-\tau )o+\tau \varsigma ,(1-\theta ) \rho + \theta q \bigr) \bigr] \\& \quad \supseteq {h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr)\\& \qquad {}\times \bigl[ \mathcal{F} \bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1-\theta )q \bigr) \mathcal{G} \bigl(\tau o+(1-\tau )\varsigma ,\theta \rho +(1- \theta )q \bigr) \\& \qquad {}+\mathcal{F} \bigl((1-\tau )o+\tau \varsigma ,\theta \rho +(1- \theta )q \bigr)\mathcal{G} \bigl((1-\tau )o+\tau \varsigma ,\theta \rho +(1- \theta )q \bigr) \\& \qquad {}+\mathcal{F} \bigl(\tau o+(1-\tau )\varsigma ,(1-\theta )\rho + \theta q \bigr)\mathcal{G} \bigl(\tau o+(1-\tau )\varsigma ,(1-\theta ) \rho + \theta q \bigr) \\& \qquad {}+\mathcal{F} \bigl((1-\tau )o+\tau \varsigma ,(1-\theta )\rho + \theta q \bigr)\mathcal{G} \bigl((1-\tau )o+\tau \varsigma ,(1-\theta ) \rho + \theta q \bigr) \bigr] \\& \qquad {}+{h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr)\\& \qquad {}\times \bigl[h_{1}(\tau )h_{1}( \theta ) \bigl[h_{2}( \tau )h_{2}(1-\theta )+h_{2}(1-\tau )h_{2}( \theta )+h_{2}(1-\tau )h_{2}(1-\theta ) \bigr] \\& \qquad {}+h_{1}(\tau )h_{1}(1-\theta ) \bigl[h_{2}(\tau )h_{2}(\theta )+h_{2}(1- \tau )h_{2}(1-\theta )+h_{2}(1-\tau )h_{2}(\theta ) \bigr] \\& \qquad {}+h_{1}(1-\tau )h_{1}(\theta ) \bigl[h_{2}(1-\tau )h_{2}(1- \theta )+h_{2}(\tau )h_{2}(\theta )+h_{2}(\tau )h_{2}(1-\theta ) \bigr] \\& \qquad {}+h_{1}(1-\tau )h_{1}(1-\theta ) \bigl[h_{2}(\tau )h_{2}( \theta )+h_{2}(1-\tau )h_{2}(\theta )+h_{2}(\tau )h_{2}(1-\theta ) \bigr]\mathcal{M}(o,\varsigma ,\rho ,q) \bigr] \\& \qquad {}+{h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr) \bigl[h_{1}(\tau )h_{1}( \theta ) \bigl[h_{2}( \tau )h_{2}(\theta )+h_{2}(1-\tau )h_{2}(1- \theta )+h_{2}(1-\tau )h_{2}(\theta ) \bigr] \\& \qquad {}+h_{1}(\tau )h_{1}(1-\theta ) \bigl[h_{2}(\tau )h_{2}(1- \theta )+h_{2}(1-\tau )h_{2}(\theta )+h_{2}(1-\tau )h_{2}(1-\theta ) \bigr] \\& \qquad {}+h_{1}(1-\tau )h_{1}(\theta ) \bigl[h_{2}(1-\tau )h_{2}( \theta )+h_{2}(\tau )h_{2}(1-\theta )+h_{2}(\tau )h_{2}(\theta ) \bigr] \\& \qquad {}+h_{1}(1-\tau )h_{1}(1-\theta )\\& \qquad {}\times \bigl[h_{2}(1-\tau )h_{2}(1- \theta )+h_{2}(\tau )h_{2}(\theta )+h_{2}(\tau )h_{2}(1-\theta ) \bigr]\mathcal{N}(o,\varsigma ,\rho ,q) \bigr] \\& \qquad {}+{h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr)\\& \qquad {}\times \bigl[h_{1}(\tau )h_{1}( \theta ) \bigl[h_{2}(1- \tau )h_{2}(1-\theta )+h_{2}(\tau )h_{2}( \theta )+h_{2}(\tau )h_{2}(1-\theta ) \bigr] \\& \qquad {}+h_{1}(\tau )h_{1}(1-\theta ) \bigl[h_{2}(1-\tau )h_{2}( \theta )+h_{2}(\tau )h_{2}(1-\theta )+h_{2}(\tau )h_{2}(\theta ) \bigr] \\& \qquad {}+h_{1}(1-\tau )h_{1}(\theta ) \bigl[h_{2}(\tau )h_{2}(1- \theta )+h_{2}(1-\tau )h_{2}(\theta )+h_{2}(1-\tau )h_{2}(1-\theta ) \bigr] \\& \qquad {}+h_{1}(1-\tau )h_{1}(1-\theta )\\& \qquad {}\times \bigl[h_{2}(\tau )h_{2}( \theta )+h_{2}(1-\tau )h_{2}(1-\theta )+h_{2}(1-\tau )h_{2}(\theta ) \bigr]\mathcal{P}(o,\varsigma ,\rho ,q) \bigr] \\& \qquad {}+{h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr) \bigl[h_{1}(\tau )h_{1}( \theta ) \bigl[h_{2}(1- \tau )h_{2}(\theta )+h_{2}(\tau )h_{2}(1- \theta )+h_{2}(\tau )h_{2}(\theta ) \bigr] \\& \qquad {}+h_{1}(\tau )h_{1}(1-\theta ) \bigl[h_{2}(1-\tau )h_{2}(1- \theta )+h_{2}(\tau )h_{2}(\theta )+h_{2}(\tau )h_{2}(1-\theta ) \bigr] \\& \qquad {}+h_{1}(1-\tau )h_{1}(\theta ) \bigl[h_{2}(\tau )h_{2}(\theta )+h_{2}(1- \tau )h_{2}(1-\theta )+h_{2}(1-\tau )h_{2}(\theta ) \bigr] \\& \qquad {}+h_{1}(1-\tau )h_{1}(1-\theta )\\& \qquad {}\times \bigl[h_{2}(\tau )h_{2}(1- \theta )+h_{2}(1-\tau )h_{2}(1-\theta )+h_{2}(1-\tau )h_{2}(\theta ) \bigr]\mathcal{Q}(o,\varsigma ,\rho ,q) \bigr]. \end{aligned}$$

Moreover, we have

$$\begin{aligned}& \frac{1}{\alpha \beta } \mathcal{F} \biggl(\frac{o+\varsigma }{2}, \frac{\rho +q}{2} \biggr)\mathcal{G} \biggl( \frac{o+\varsigma }{2}, \frac{\rho +q}{2} \biggr) \\& \quad \supseteq \frac{\Gamma (\alpha )\Gamma (\beta ){h_{1}}^{2}(\frac{1}{2}){h_{2}}^{2}(\frac{1}{2})}{(\varsigma -o)^{\alpha }(q-\rho )^{\beta }}\\& \qquad {}\times \bigl[\mathfrak{J}_{o^{+},\rho ^{+}}^{\alpha ,\beta } \mathcal{F}( \varsigma ,q)+\mathfrak{J}_{o^{+},q^{-}}^{\alpha ,\beta }\mathcal{F}( \varsigma ,\rho )+\mathfrak{J}_{\varsigma ^{-},\rho ^{+}}^{\alpha , \beta }\mathcal{F}(o,q)+ \mathfrak{J}_{\varsigma ^{-},q^{-}}^{\alpha , \beta }\mathcal{F}(o,\rho ) \bigr] \\& \qquad {}+2{h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr) \mathcal{M}(o, \varsigma ,\rho ,q) \\& \qquad {}\times\int _{0}^{1}\tau ^{\alpha -1}\,d\tau \int _{0}^{1} \theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta ) \bigl[h_{2}(\tau )h_{2}(1- \theta ) \\& \qquad {}+h_{2}(1-\tau )h_{2}(\theta )+h_{2}(1-\tau )h_{2}(1-\theta ) \bigr]\\& \qquad {}+h_{1}(\tau )h_{1}(1- \theta ) \bigl[h_{2}( \tau )h_{2}(\theta )+h_{2}(1-\tau )h_{2}(1- \theta )+h_{2}(1-\tau )h_{2}(\theta ) \bigr] \bigr]\,d\theta \\& \qquad {}+2{h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr) \mathcal{N}(o, \varsigma ,\rho ,q) \\& \qquad {}\times\int _{0}^{1}\tau ^{\alpha -1}\,d\tau \int _{0}^{1} \theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta ) \bigl[h_{2}(\tau )h_{2}( \theta ) \\& \qquad {}+h_{2}(1-\tau )h_{2}(1-\theta )+h_{2}(1-\tau )h_{2}(\theta ) \bigr]\\& \qquad {}+h_{1}(\tau )h_{1}(1- \theta ) \bigl[h_{2}( \tau )h_{2}(1-\theta )+h_{2}(1-\tau )h_{2}( \theta )+h_{2}(1-\tau )h_{2}(1-\theta ) \bigr] \bigr]\,d \theta \\& \qquad {}+2{h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr) \mathcal{P}(o, \varsigma ,\rho ,q) \\& \qquad {}\times\int _{0}^{1}\tau ^{\alpha -1}\,d\tau \int _{0}^{1} \theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta ) \bigl[h_{2}(1- \tau )h_{2}(1-\theta ) \\& \qquad {}+h_{2}(\tau )h_{2}(\theta )+h_{2}(\tau )h_{2}(1-\theta ) \bigr]\\& \qquad {}+h_{1}(\tau )h_{1}(1- \theta ) \bigl[h_{2}(1- \tau )h_{2}(\theta )+h_{2}(\tau )h_{2}(1- \theta )+h_{2}(\tau )h_{2}(\theta ) \bigr] \bigr]\,d\theta \\& \qquad {}+2{h_{1}}^{2}\biggl(\frac{1}{2} \biggr){h_{2}}^{2}\biggl(\frac{1}{2}\biggr) \mathcal{Q}(o, \varsigma ,\rho ,q) \\& \qquad {}\times\int _{0}^{1}\tau ^{\alpha -1}\,d\tau \int _{0}^{1} \theta ^{\beta -1} \bigl[h_{1}(\tau )h_{1}(\theta ) \bigl[h_{2}(1- \tau )h_{2}(\theta ) \\& \qquad {}+h_{2}(\tau )h_{2}(1-\theta )+h_{2}(\tau )h_{2}(\theta ) \bigr]\\& \qquad {}+h_{1}(\tau )h_{1}(1- \theta ) \bigl[h_{2}(1- \tau )h_{2}(1-\theta )+h_{2}(\tau )h_{2}( \theta )+h_{2}(\tau )h_{2}(1-\theta ) \bigr] \bigr]\,d\theta , \end{aligned}$$

which rearranges to the required result. □

Remark 3.9

If \(\alpha =\beta =1\) and \(h(\theta )=\theta \), then we get Theorem 9 of [26]. If \(\underline{\mathcal{F}}=\overline{\mathcal{F}}\), \(h(\theta )=\theta \), and \(\alpha =\beta =1\), then we get Theorem 5 of [35].

Conclusion

In this paper, we proved some new Hermite–Hadamard-type inequalities for coordinated h-convex interval-valued functions via Riemann–Liouville-type fractional integrals. The results generalize the previous results given in [2527, 33, 35]. Moreover, in the future investigation, these results may be extended for different kinds of convexities and fractional integrals.

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Acknowledgements

The authors thank the anonymous referees for invaluable comments and insightful suggestions, which improved the presentation of this manuscript.

Funding

This work was supported in part by the National Key Research and Development Program of China (2018YFC1508100), Key Projects of Educational Commission of Hubei Province of China (D20192501)and the Natural Science Foundation of Jiangsu Province (BK20180500).

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Correspondence to Guoju Ye.

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Shi, F., Ye, G., Zhao, D. et al. Some fractional Hermite–Hadamard-type inequalities for interval-valued coordinated functions. Adv Differ Equ 2021, 32 (2021). https://doi.org/10.1186/s13662-020-03200-z

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MSC

  • 26D15
  • 26E25
  • 26A33

Keywords

  • Hermite–Hadamard-type inequalities
  • Interval-valued coordinated functions
  • Riemann–Liouville-type fractional integrals
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