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

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 where f : I → R is a convex function on the closed bounded interval I of R, and o, ς ∈ I with o < ς . 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 [1][2][3][4][5][6]. 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-Hadamardtype 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 [7][8][9][10][11][12][13][14][15][16][17][18][19].
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 intervalvalued 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-Hadamardtype 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 hconvex 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 R I the set of closed bounded intervals of R and by R + and R + I the sets of positive real numbers and positive intervals, respectively. We also denote = [o, ς] × [ρ, q]. For more notions on interval-valued functions, see [28,29].
We denote the set of all h-convex interval-valued functions by SX(h, [o, ς], R + I ).
where α > 0, and is the gamma function.
The Riemann-Liouville-type fractional integrals of interval-valued coordinated functions F(t, s) are given as follows.
Proof Using Theorem 2.7 and F ∈ SX(ch, , R + I ), we get which gives the second and third inequalities in (3.6).
Using the first inequality in (2.1), we get 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 which gives the last inequality in (3.6). This completes the proof.

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 [25-27, 33, 35]. Moreover, in the future investigation, these results may be extended for different kinds of convexities and fractional integrals.