Matrix Inequalities between $f(A)\sigma f(B)$ and $A\sigma B$

TL;DR

This paper establishes new matrix inequalities involving convex and concave functions of positive definite matrices under matrix means, improving known subadditivity inequalities and generalizing Minkowski's determinant inequality.

Contribution

It introduces novel bounds for $f(A)\sigma f(B)$ in terms of $A \sigma B$ for convex and concave functions, extending matrix inequality theory.

Findings

Derived bounds for $f(A)\sigma f(B)$ using eigenvalue-based constants.

Improved subadditivity inequalities for unitarily invariant norms.

Generalized Minkowski's determinant inequality.

Abstract

Let $A$ and $ B$ be $n\times n$ positive definite complex matrices, let $\sigma$ be a matrix mean, and let $f : [0,\infty)\to [0,\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\prime}(0)(A \sigma B)\leq \frac{f(m)}{m}(A\sigma B)\leq f(A)\sigma f(B)\leq \frac{f(M)}{M}(A\sigma B)\leq f^{\prime}(M)(A\sigma B),$$ where $m$ represents the smallest eigenvalues of $A$ and $B$ and $M$ represents the largest eigenvalues of $A$ and $B$. If $f$ is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if $f(x)/x$ is increasing, then $$|||f(A)+f(B)|||\leq\frac{f(M)}{M} |||A+B|||\leq |||f(A+B)|||$$ holds for all $A$ and $B$ with $M\leq A+B$. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski's determinant inequality.equality.