Norm Inequalities via Convex and Log-Convex Functions

Abstract

In this paper, we study the norm and skew angular distances in a normed space X, 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 a, b∈ X are non-zero and if p, q> 0 are such that 1p+1q=1, then 2λ(pr∥a∥r+qr∥b∥r2-∥pa+qb2∥r)≤pr-1∥a∥r+qr-1∥b∥r-∥a+b∥r≤2μ(pr∥a∥r+qr∥b∥r2-∥pa+qb2∥r),where r≥ 1 , λ= min { 1 / p, 1 / q} and μ= max { 1 / p, 1 / q}. 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 ‖ AtXB1-t‖ , where A, B are positive definite matrices, X is an arbitrary matrix, ‖ · ‖ is a unitarily invariant norm and 0 ≤ t≤ 1. Many other results involving matrix and scalar log-convex functions will be presented too.