On Hadamard Type Integral Inequalities For Nonconvex Functions

Convexity plays a central and fundamental role in mathematical finance, economics, engineering, management sciences and optimizastion theory. In recent years, several extensions and generalizations have been considered for classical convexity. A significant generalization of convex functions is that of ϕ-convex functions introduced by Noor in [3]. In [3] and [6], the authors have studied the basic properties of the ϕ-convex functions. It is well-know that the ϕ-convex functions and ϕ-sets may not be convex functions and convex sets. This class of nonconvex functions include the classical convex functions and its various classes as special cases. For some recent results related to this nonconvex functions, see the papers [3]-[6]. In this article, using functions whose derivatives absolute values are ϕ-convex and quasi-ϕ-convex, we obtained new inequalities releted to the right and left side of Hermite-Hadamard inequality. In particular if ϕ=0 is taken as, our results obtained reduce to the Hermite-Hadamard type inequality for classical convex functions.