Hermite–Hadamard-type inequalities for interval-valued preinvex functions via Riemann–Liouville fractional integrals

*Correspondence: abhamdi@qu.edu.qa 2Department of Mathematics, Statistics and Physics College of Arts and Sciences, Qatar University, P. O. Box 2713, Doha, Qatar Full list of author information is available at the end of the article Abstract In this paper, we introduce (h1,h2)-preinvex interval-valued function and establish the Hermite–Hadamard inequality for preinvex interval-valued functions by using interval-valued Riemann–Liouville fractional integrals. We obtain Hermite–Hadamard-type inequalities for the product of two interval-valued functions. Further, some examples are given to confirm our theoretical results.

al. [1] discussed dynamic inequalities on time scales such as Young's inequality, Hölder's inequality, Minkoswki's inequality, Jensen's inequality, Steffensen's inequality, Hermite-Hadamard inequality, Čebyšv's inequality, and Opial type inequality. In 2010, Srivastava et al. [24] established some general weighted Opial type inequalities on time scales. Further, Srivastava et al. [25] presented some extensions and generalizations of Maroni's inequality to hold true on time. Wei et al. [29] established local fractional integral analogue of Anderson's inequality on fractal space under some suitable conditions and also showed that the local fractional integral inequality on fractal space is a new generalization of the classical Anderson's inequality. Further, Tunç et al. [27] established an identity for local fractional integrals and derived several generalizations of the celebrated Steffensen's inequality associated with local fractional integrals. For more details, we can refer to [1,26] and the references therein.
Bhurjee and Panda [3] defined the interval-valued function in the parametric form and developed a methodology to study the existence of the solution of a general interval optimization problem. Lupulescu [14] introduced the differentiability and integrability for the interval-valued functions on time scales by using the concept of the generalized Hukuhara difference. In 2015, Cano et al. [7] proposed a new Ostrowski type inequalities for gHdifferentiable interval-valued functions and obtained generalization of the class of real functions which is not necessarily differentiable. Cano et al. [7] obtained error bounds to quadrature rules for gH-differentiable interval-valued functions. Further, Roy and Panda [22] introduced the concept of μ-monotonic property of interval-valued function in the higher dimension and derived some results by using generalized Hukuhara differentiability. For more details of interval-valued functions, we refer to [4,5,11,13,15,22] and the references therein.
Recently, An et al. [2] introduced (h 1 , h 2 )-convex interval-valued function and obtained some interval Hermite-Hadamard type inequalities. Further, Budak et al. [6] established the Hermite-Hadamard inequality for the convex interval-valued function and for the product of two convex interval-valued functions.
Motivated by the above works and ideas, we introduce the concept of (h 1 , h 2 )-preinvex interval-valued function and establish the Hermite-Hadamard inequality for preinvex interval-valued functions and for the product of two preinvex interval-valued functions via interval-valued Riemann-Liouville fractional integrals. Also, we give some examples in the support of our theory.

Preliminaries
In this section, we mention some definitions and related results required for this manuscript.

Interval arithmetic
The rules for interval addition, subtraction, product, and quotient [18] are It is easy to see that X.Y is again an interval, whose end points can be computed from The reciprocal of an interval is as follows: If X is an interval not containing the number 0, then where 1/y is defined by (1). Scalar multiplication of the interval X is defined by where λ ∈ R. Let R I , R + I , and R -I be the sets of all closed intervals of R, sets of all positive closed intervals of R, and sets of all negative closed intervals of R, respectively. Now, we discuss some algebraic properties of interval arithmetic [18].

Integral of interval-valued functions:
A function ξ is said to be an interval-valued function of δ on [c, d] if it assigns a nonempty interval to each δ ∈ [c, d] where ξ and ξ are real-valued functions. A partition of [c, d] is any finite ordered subset P having the form The mesh of a partition P is defined by . . , n, and we define the sum .
and is denoted by . and respectively. Here, (α) is the gamma function and J 0 where (α) is the gamma function.
where (α) is the gamma function.

Definition 5 ([30])
The set A ⊆ R n is said to be invex with respect to a vector function η : It is well known that every convex set is invex with respect to η(x, y) = xy but not conversely.

Definition 6 ([30])
The function ξ on the invex set A is said to be preinvex with respect to η if It is well known that every convex function is preinvex with respect to η(x, y) = xy but not conversely.

Main results
In this section, first, we give the definition of interval-valued h-preinvex function and discuss some special cases of interval-valued h-preinvex functions.   + η(d, c). c + η(d, c)] and η satisfies Condition C and α > 0, then we have Proof Since ξ is a preinvex interval-valued function, we have c) and Condition C in (4), we get This implies Multiplying by δ α-1 , α > 0 on both sides in (5), we have Integrating the above inequality on [0, 1], we get Applying Theorem 1 in the above relation, we get This implies Similarly, Using (8), (9), and (10) in (7), we have Now, we prove the second pair of inequalities.