The intention of this article is to investigate the most important inequalities of m-convex functions without using their derivatives. The article also provides a brief survey of general properties of m-convex functions. | Turk J Math (2017) 41: 625 – 635 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article The most important inequalities of m-convex functions 1 ´ 1 , Merve AVCI ARDIC Zlatko PAVIC ¸ 2,∗ Department of Mathematics, Mechanical Engineering Faculty in Slavonski Brod, University of Osijek, Slavonski Brod, Croatia 2 Department of Mathematics, Faculty of Science and Arts, Adıyaman University, Adıyaman, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: The intention of this article is to investigate the most important inequalities of m -convex functions without using their derivatives. The article also provides a brief survey of general properties of m -convex functions. Key words: m -Convex function, Jensen inequality, Fej´er inequality, Hermite–Hadamard inequality 1. Introduction Let [a, b] ⊂ R be an interval where a b . It is suitable to use the convex hull of points a and b as the set conv{a, b} = {(1 − t)a + tb : t ∈ [0, 1]} , including the case a = b. If the interval [a, b] contains the zero, then the product mc belongs to [a, b] for every c ∈ [a, b] and m ∈ [0, 1]. Thus, if points x, y ∈ [a, b] and coefficients t, m ∈ [0, 1], then the convex combination (1 − t)x + tmy of points x and my belongs to the convex hull conv{x, my} ⊆ [a, b]. Any interval I ⊆ R containing the zero possesses the above properties. Throughout the paper, we will use proper intervals (with the nonempty interior) of real numbers containing the zero. Definition 1 Let I ⊆ R be an interval containing the zero, and let m ∈ (0, 1] be a number. A function f : I → R is said to be m -convex if the inequality ( ) f (1 − t)x + tmy ≤ (1 − t)f (x) + tmf (y) () holds for every pair of points x, y ∈ I and every coefficient t ∈ [0, 1]. We point out the following note relating to the above definition. Using the point x = my or the coefficient t = 1 in formula (), we get the
