Around Jensen’s inequality for strongly convex functions

Abstract

In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen-Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen's operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that iffor each positive operator A and x ∈ H with x = 1.2010 Mathematics Subject Classification. Primary 47A63, 26B25. Secondary 46L05.

A complementary inequality to the information monotonicity for Tsallis relative operator entropy

“…Proof. We follow a similar path as in the proof of Theorem 3.3 in [12]. Due to relation (2.3), we have…”

Majorization theorems for strongly convex functions

“…Convex functions were proposed by Jensen over 100 years ago. Over the past few years, many generalizations and extensions have been made for the convexity, for example, quasi-convexity [19], strong convexity [20,21], approximate convexity [22], logarithmical convexity [23], midconvexity [24], pseudo-convexity [25], h-convexity [26], deltaconvexity [27], s-convexity [28], preinvexity [29], GA-convexity [30], GG-convexity [31], coordinate strong convexity [32], and Schur convexity [33][34][35][36][37][38][39][40][41][42][43][44]. In particular, many remarkable inequalities can be found in the literature via the convexity theory.…”

Integral Inequalities Involving Strongly Convex Functions

“…For more recent results related to strongly convex function and Jensen type inequalities we recommend [22,[29][30][31][32][33][34].…”