Some Properties of h-MN-Convexity and Jensen’s Type Inequalities

In this work, we introduce the class of h-MN-convex functions by generalizing the concept of MN-convexity and combining it with h-convexity. Namely, let M : [0, 1] → [a, b] be a Mean function given by M (t) = M (t; a, b); where by M (t; a, b) we mean one of the following functions: At (a, b) := (1 − t) a + tb, Gt (a, b) = a1−tbt and Ht (a, b) := ab ta+( 1−t )b =  1 A t ( 1 a , 1 b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVCI8FfYJH8 YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=J Hqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaaca qabeaadaqaaqaafaGcbaaeaaaaaaaaa8qadaWcaaWdaeaapeGaamyy aiaadkgaa8aabaWdbiaadshacaWGHbGaey4kaSYaaeWaa8aabaWdbi aaigdacqGHsislcaWG0baacaGLOaGaayzkaaGaamOyaaaacqGH9aqp caGGGcWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaadgeapaWaaSbaaS qaa8qacaWG0baapaqabaGcpeWaaeWaa8aabaWdbmaalaaapaqaa8qa caaIXaaapaqaa8qacaWGHbaaaiaacYcadaWcaaWdaeaapeGaaGymaa WdaeaapeGaamOyaaaaaiaawIcacaGLPaaaaaaaaa@54E0@ with the property that M (0; a, b) = a and M (1; a, b) = b. Let I, J be two intervals subset of (0, ∞) such that (0, 1) ⊆ J and [a, b] ⊆ I. Consider a non-negative function h : J → (0, ∞), a function f : I → (0, ∞) is said to be h-MN-convex (concave) if the inequality f (M (t; x, y)) ≤ (≥) N (h(t); f (x), f (y)), holds for all x, y ∈ I and t ∈ [0, 1]. In this way, nine classes of h-MN-convex functions are established, and therefore some analytic properties for each class of functions are explored and investigated. Characterizations of each type are given. Various Jensen’s type inequalities and their converses are proved.