Norm Inequalities via Convex and Log-Convex Functions

Abstract

In this paper, we study the norm and skew angular distances in a normed space , where convex functions are used to obtain refinements and reverses of some outstanding results in the literature. For example, in this regard, we show that if are non-zero and if are such that , then

where , and . Then we explain how this result extends some known results in the literature. Many other related results will be also shown. Then, with the theme of convexity, we employ a log-convex approach on certain matrix functions to obtain improvements and new sights of some matrix inequalities, including possible bounds of where A, B are positive definite matrices, X is an arbitrary matrix, is a unitarily invariant norm and Many other results involving matrix and scalar log-convex functions will be presented too.