On the Hermite–Hadamard inequalities for interval-valued coordinated convex functions

In this work, we introduce the notion of interval-valued coordinated convexity and demonstrate Hermite–Hadamard type inequalities for interval-valued convex functions on the co-ordinates in a rectangle from the plane. Moreover, we prove Hermite–Hadamard inequalities for the product of interval-valued convex functions on coordinates. Our results generalize several other well-known inequalities given in the existing literature on this subject.


Introduction
The classical Hermite-Hadamard inequality is one of the most well-established inequalities in the theory of convex functions with geometrical interpretation, and it has many applications. The Hermite-Hadamard inequality states that, if f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b, then (1.1) Both inequalities in (1.1) hold in the reversed direction if f is concave. We note that Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity, and it is implied easily from Jensen's inequality. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years, and a remarkable variety of refinements and generalizations have been studied. In [7], Dragomir demonstrated the subsequent inequality of Hadamard type for coordinated convex functions.
For more results related to (1.2), we refer the readers to [1,9,15] and the references therein.
On the other hand, interval analysis is a notable case of set-valued analysis, which is the discussion of sets in the spirit of mathematical analysis and general topology. It was introduced as an attempt to handle the interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. An old example of an interval enclosure is Archimede's method, which is related to computing the circumference of a circle. In 1966, the first book related to interval analysis was given by Moore, who is known as the first user of intervals in computational mathematics, see [11]. After his book, several scientists started to investigate the theory and application of interval arithmetic. Nowadays, because of its applications, interval analysis is a useful tool in various areas which are interested intensely in uncertain data. You can see applications in computer graphics, experimental and computational physics, error analysis, robotics, and many others.

Preliminaries and known results
In this section, we review some basic definitions, results, notions, and properties which are used throughout the paper. The set of all closed intervals of R, the sets of all closed positive intervals of R, and closed negative intervals of R are denoted by R I , R + I , R -I , respectively. The Hausdorff distance between [X, X] and [Y , Y ] is defined as The metric space (R I , d) is a complete metric space. For more in-depth notations on interval-valued functions, see [12,19].
In [11], Moore gave the notion of the Riemann integral for interval-valued functions. The sets of all Riemann integrable interval-valued functions and real-valued functions on [a, b] are denoted by IR ([a,b]) and R ([a,b]) , respectively. The following theorem gives a relation between (IR)-integrable functions and Riemann integrable (R-integrable) functions (see, [12, p. 131]).

Theorem 2 Let F : [a, b] → R I be an interval-valued function with the property that F(t) = [F(t), F(t)]. F ∈ IR ([a,b]) if and only if F(t), F(t) ∈ R ([a,b]) and
In [19,21], Zhao et al. introduced a kind of convex interval-valued function as follows.
With SX(h, [a, b], R + I ), we will show the set of all h-convex interval-valued functions.
The usual notion of convex interval-valued function matches a relation (2.1) with h(t) = t (see [18]). Moreover, if we take h(t) = t s in (2.1), then Definition 1 gives the s-convex interval-valued function defined by Breckner (see [2]).
In [19], Zhao et al. obtained the following Hermite-Hadamard inequality for intervalvalued functions by using h-convexity.
2) reduces to the following result: which was obtained by Sadowska in [18].
2) reduces to the following result: which was obtained by Osuna-Gómez et al. in [14].
Remark 2 If h(t) = t, then (2.4) reduces to the following result: then (2.5) reduces to the following result: We call S(F, P, δ, ) an integral sum of F associated with P ∈ P(δ, ). Now, we review the concepts and notations of interval-valued double integral given by Zhao et al. in [20]. Definition 2 A function F : → R + I is said to be interval-valued coordinated convex function if the following inequality holds: for all (x, y), (u, w) ∈ and s, t ∈ [0, 1].

Lemma 1 A function F : → R + I is an interval-valued convex on coordinates if and only if there exist two functions F
Proof The proof of this lemma follows immediately by the definition of interval-valued coordinated convex function.
It is easy to prove that an interval-valued convex function is interval-valued coordinated convex, but the converse may not be true. For this, we can see the following example. In what follows, without causing confusion, we will delete the notations of (R), (IR), and (ID). We start with the following theorem.

Theorem 7 If F : → R + I is an interval-valued coordinated convex function on such that F(t) = [F(t), F(t)], then the following inequalities hold:
which can be written as Integrating (4.2) with respect to x over [a, b] and dividing both sides by ba, we have By adding (4.3) and (4.4) and using Theorem 2, we have the second and third inequality in (4.1). From (2.3) we also have By adding (4.5) and (4.6) and using Theorem 2, we have the first inequality in (4.1). In the end, again from (2.2) and Theorem 2, we have and the proof is completed.
Remark 4 If F = F, then Theorem 7 reduces to Theorem 1.
Proof Since F and G are interval-valued coordinated convex functions on , therefore which can be written as Integrating the above inequality with respect to x over [a, b] and dividing both sides by ba, we have (4.8) Now, using inequality (2.6) for each integral on the right-hand side of (4.8), we have c) .   N(a, b, c, d), (4.13) where P (a, b, c, d), M(a, b, c, d), and N(a, b, c, d) are defined in Theorem 8.
Proof Since F and G are interval-valued coordinated convex functions, from (2.7) we have and Adding (4.14) and (4.15), then multiplying both sides of the resultant one by 2, we get  F(a, c)G(b, c) + F(b, c)G(a, c) , Using (4.17)-(4.24) in (4.16), we have