Some properties of h-MN-convexity and Jensen's type inequalities. Issue 8 (17th November 2019)

• Record Type: Journal Article
• Authors: Alomari, Mohammad W.
• Is Part Of: Journal of interdisciplinary mathematics. Volume 22:Issue 8(2019)
• Page Start: 1349
• Page End: 1395
• Publication Date: 2019-11-17
• Subjects: (2000) 26A51 -- 26A15 -- 26E60; h-convex function -- Means -- Jensen inequality; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.1080/09720502.2019.1698402
• Languages: English
• ISSNs: 0972-0502

Abstract

Abstract: 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 I, J be two intervals subset of (0, ∞) such that (0, 1) ⊆ J and [ a, b ] ⊆ I . Consider a non-negative function h : (0, ∞) → (0, ∞) and let M : [0, 1] → [a, b] (0 < a < b) be a Mean function given by M ( t ) = M ( h ( t ); a, b ); where by M ( h ( t ); a, b ) we mean one of the following functions: Ah ( a, b ) ≔ h (1 – t ) a + h ( t ) b, Gh ( a, b ) = a h (1– t ) b h ( t ) and ; with the property that M ( h (0); a, b ) and M ( h (1); a, b ) = b . A function f : I → (0, ∞) is said to be h -MN-convex (concave) if the inequality holds for all x, y ∈ I and t ∈ [0, 1], where M and N are two mean functions. In this way, nine classes of h -MN-convex functions are established and some of their analytic properties are explored and investigated. Characterizations of each type are given. Various Jensen's type inequalities and their converses are proved.