On the logarithmic mean of accretive matrices

Abstract

In this paper, we define the logarithmic mean of accretive matrices and study its basic properties. Among other results, we show that if $A, B$ are accretive matrices, then $$\Re L(A, B)\ge L(\Re A, \Re B),$$
where $L(A, B)$ is the logarithmic mean of $A$ and $B$, and $\Re A$ means the real part of $A$. This complements a recent result of Lin and Sun \cite{LS17}.