Extensions of interpolation between the arithmetic-geometric mean inequality for matrices

TL;DR

This paper extends interpolation inequalities related to the arithmetic-geometric mean for matrices, providing new bounds involving matrix functions and unitarily invariant norms.

Contribution

It introduces novel matrix inequalities that generalize existing bounds using functions and parameters, expanding the theoretical framework of matrix mean inequalities.

Findings

Established new inequalities involving matrix functions and norms.

Generalized interpolation inequalities for matrices with parameterized functions.

Provided bounds applicable to unitarily invariant norms.

Abstract

In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if $A, B, X$ are $n\times n$ matrices, then \begin{align*} \|AXB^*\|^2\leq\|f_1(A^*A)Xg_1(B^*B)\|\,\|f_2(A^*A)Xg_2(B^*B)\|, \end{align*} where $f_1,f_2,g_1,g_2$ are non-negative continues functions such that $f_1(t)f_2(t)=t$ and $g_1(t)g_2(t)=t\,\,(t\geq0)$. We also obtain the inequality \begin{align*} \left|\left|\left|AB^*\right|\right|\right|^2\nonumber&\leq \left|\left|\left|p(A^*A)^{\frac{m}{p}}+ (1-p)(B^*B)^{\frac{s}{1-p}}\right|\right|\right|\,\left|\left|\left|(1-p)(A^*A)^{\frac{n}{1-p}}+ p(B^*B)^{\frac{t}{p}}\right|\right|\right|, \end{align*} in which $m,n,s,t$ are real numbers such that $m+n=s+t=1$, $|||\cdot|||$ is an arbitrary unitarily invariant norm and $p\in[0,1]$.