Hermite-Hadamard-Type Inequalities for r-Preinvex Functions

Several interesting generalizations and extensions of classical convexity have been studied and investigated in recent years. Hanson [1] introduced the invex functions as a generalization of convex functions. Later, subsequent works inspired from Hanson’s result have greatly found the role and applications of invexity in nonlinear optimization and other branches of pure and applied sciences. The basic properties and role of preinvex functions in optimization, equilibrium problems and variational inequalities were studied by Noor [2, 3] and Weir and Mond [4]. The well known Hermite-Hadamard inequality has been extensively investigated for the convex functions and their variant forms. Noor [5] and Pachpatte [6] studied these inequalities and related types for preinvex and log-preinvex functions. The Hermite-Hadamard inequality was investigated in [7] to an r-convex positive function which is defined on an interval [a, b]. Certain refinements of the Hadamard inequality for r-convex functions were studied in [8, 9]. Recently, Zabandan et al. [10] extended and refined the work of [8]. Bessenyei [11] studied Hermite-Hadamard-type inequalities for generalized 3-convex functions. In this paper, we introduce the r-preinvex functions and establishHermiteHadamard inequality for such functions by using the method of [10].


Introduction
Several interesting generalizations and extensions of classical convexity have been studied and investigated in recent years.
Hanson [1] introduced the invex functions as a generalization of convex functions.Later, subsequent works inspired from Hanson's result have greatly found the role and applications of invexity in nonlinear optimization and other branches of pure and applied sciences.The basic properties and role of preinvex functions in optimization, equilibrium problems and variational inequalities were studied by Noor [2,3] and Weir and Mond [4].The well known Hermite-Hadamard inequality has been extensively investigated for the convex functions and their variant forms.Noor [5] and Pachpatte [6] studied these inequalities and related types for preinvex and log-preinvex functions.
The Hermite-Hadamard inequality was investigated in [7] to an -convex positive function which is defined on an interval [, ].Certain refinements of the Hadamard inequality for -convex functions were studied in [8,9].Recently, Zabandan et al. [10] extended and refined the work of [8].Bessenyei [11] studied Hermite-Hadamard-type inequalities for generalized 3-convex functions.In this paper, we introduce the -preinvex functions and establish Hermite-Hadamard inequality for such functions by using the method of [10].

Preliminaries
Let  :  → R and (⋅, ⋅) :  ×  → R be continuous functions, where  ⊂ R  is a nonempty closed set.We use the notations, ⟨⋅, ⋅⟩ and ‖ ⋅ ‖, for inner product and norm, respectively.We require the following well known concepts and results which are essential in our investigations.
Definition 1 (see [2,4]).Let  ∈ .Then, the set  is said to be invex at  ∈  with respect to (⋅, ⋅) if is said to be invex set with respect to (⋅, ⋅) if it is invex at every  ∈ .The invex set  is also called a  connected set.Geometrically, Definition 1 says that there is a path starting from the point  which is contained in .The point V should not be one of the end points of the path in general, see [12].This observation plays a key role in our study.If we require that V should be an end point of the path for every pair of points , V ∈ , then (V, ) = V − , and consequently, invexity reduces to convexity.Thus, every convex set is also an invex set with respect to (V, ) = V − , but the converse is not necessarily true; see [4,13] and the references therein.For the sake of simplicity, we assume that  = [,  + (, )], unless otherwise specified.Definition 2 (see [4]).The function  on the invex set  is said to be preinvex with respect to  if Note that every convex function is a preinvex function, but the converse is not true.For example, the function () = The concepts of the invex and preinvex functions have played very important roles in the development of generalized convex programming, see [14][15][16][17].For more characterizations and applications of invex and preinvex functions, we refer to [3,15,[18][19][20][21][22][23][24][25].
Antczak [26,27] introduced and studied the concept of -invex and -preinvex functions.Here, we define the following.Definition 3. A positive function  on the invex set  is said to be -preinvex with respect to  if, for each , V ∈ ,  ∈ [0, 1]: Note that 0-preinvex functions are logarithmic preinvex and 1-preinvex functions are classical preinvex functions.It should be noted The well known Hermite-Hadamard inequality for a convex function defined on the interval [, ] is given by see [6,28,29].Hermite-Hadamard inequalities for log-convex functions were proved by Dragomir and Mond [30].Pachpatte [6,17] also gave some other refinements of these inequalities related to differentiable log-convex functions.Noor [5] proved the following Hermite-Hadamard inequalities for the preinvex and log-preinvex functions, respectively.
Note that for  = 1, in Theorem 6, we have the same inequality again as in Corollary 5.
As a special case of Theorem 7, we deduce the following result.