257 Sigma J Eng & Nat Sci 10 (3), 2019, 257-264 Research Article SOME PROPERTIES OF (, )-PREINVEX FUNCTIONS AND HERMITE HADAMARD INEQUALITY Erdal ÜNLÜYOL* 1 , Mustafa KARADENİZ 2 1 Ordu University, Department of Mathematics, ORDU; ORCID:0000-0003-3465-6473 2 Ordu University, Department of Mathematics, ORDU Received: 15.04.2019 Revised: 07.05.2019 Accepted: 19.05.2019 ABSTRACT In this paper, firstly it is defined a new class of preinvex, namely, (h, m)-preinvex. Secondly it is obtained some algebraic properties of this class, i.e. sum, multiple etc. Finally it is proved the Hermite-Hadamard Type Inequality for (h, m) −convex and established some new inequalities. Keywords: Convex, Hermite-Hadamard, preinvex, m-preinvex, ℎ-preinvex, (ℎ, )-preinvex. MSC 2010: 26D10, 26D15. 1. INTRODUCTION Invex functions theory was introduced by Hanson [1]. Then Weir and Mond [2] defined the preinvex function. They applied the preinvex function to the establishment of the sufficient optimality conditions and duality in nonlinear programming. After that Noor [3] proved the Hermite-Hadamard inequality for preinvex and log-preinvex functions. Preinvex functions are an important generalization of convex functions. And if you want to learn more details and resources for invexity and prequasiinvex etc. you can see [4, 6], and reference therein. Now let we give some basic definitions and theorems. Definition 1 : A function :  ⊆ ℝ → ℝ is said to be convex if ( + (1 − )) ⩽ () + (1 − )() holds for every ,  ∈  and  ∈ [0,1]. Definition 2 : The following celebrated double inequality ( + 2 )⩽ 1 − ∫   () ⩽ ()+() 2 (1.1) holds for convex functions and is well-known in the literature as the Hermite-Hadamard inequality. Both the inequalities in (1.1) hold in reversed direction if f is concave. The inequality (1.1) has been a subject of extensive research since its discovery and a number of paper have been written providing noteworthy extensions, generalizations and refinements. * Corresponding Author: e-mail: eunluyol@yahoo.com, tel: (452) 226 52 00 / 1822 Publications Prepared for the Sigma Journal of Engineering and Natural Sciences 2019 International Conference on Applied Analysis and Mathematical Modeling Special Issue was published by reviewing extended papers