The Schur Multiplicative and Harmonic Convexities for Three Classes of Symmetric Functions

Abstract

We investigate the Schur harmonic convexity for two classes of symmetric functions and the Schur multiplicative convexity for a class of symmetric functions by using a new method and generalizing previous result. As applications, we establish some inequalities by use of the theory of majorization, in particular, and we give some new geometric inequalities in the n-dimensional space.

Generalization of Favard’s and Berwald’s Inequalities for Strongly Convex Functions

“…If Ψ is strictly strongly convex function and β = ζ, then the strict inequalities hold in (14) and (15) and their reverse cases.…”

“…These generalizations and extensions in the theory of inequalities have made precious contributions in different areas of mathematics. In this point of view, the new generalized concepts are quasiconvex [17], ϕ-convex [19], λ-convex [20], approximately convex [23], midconvex functions [24], pseudo-convex [29], strongly convex [31], logarithmically convex [35], h-convex [39], delta-convex [36], Schur convexity [14][15][16] and others [1-3, 10-13, 26, 34, 38, 40-44, 46].…”