New inequalities for (p,h)-convex functions for τ-measurable operators

Abstract

The main goal of this article is to present new inequalities for (p, h)-convex and (p,h) log-convex functions for a non-negative super-multiplicative and super-additive function h. Our first main result will be h? (v/?) ? (h(1 ? v) f (a) + h(v) f (b))? ? f ? [((1 ? v)ap + vbp) 1 p] / (h(1 ? ?) f (a) + h(?) f (b))? ? f ? [((1 ? ?)ap + ?bp)1p] ? h? (1 ? v/1 ? ?), for the positive (p, h)-convex function f, when ? ? 1, p ? R\{0} and 0 ? v ? ? ? 1. This gives a generalization of an important result due to M. Sababheh [Linear Algebra Appl. 506 (2016), 588-602]. As applications of our results, we present many inequalities for the trace, and the symmetric norms for ?-measurable operators.