On the weighted Geometric mean of accretive matrices

TL;DR

This paper introduces new inequalities for accretive matrices involving weighted geometric means and powers, extending known results from positive matrices to a broader class of accretive matrices.

Contribution

It develops novel inequalities for accretive matrices involving powers and geometric means for parameters outside the standard range, broadening the theoretical framework.

Findings

Established inequalities for $A^r$ and $A\sharp_rB$ with $r\in (-1,0)\cup (1,2)$

Extended known results from positive matrices to accretive matrices

Provided accretive matrix versions of classical inequalities

Abstract

In this paper, we discuss new inequalities for accretive matrices through non standard domains. In particular, we present several relations for $A^r$ and $A\sharp_rB$, when $A,B$ are accretive and $r\in (-1,0)\cup (1,2).$ This complements the well established discussion of such quantities for accretive matrices when $r\in [0,1],$ and provides accretive versions of known results for positive matrices.