Further Refinements of Zhan's Inequality for Unitarily Invariant Norms

Abstract

$\newcommand{\UIN}{|\kern-1pt|kern-1pt|}$ In this paper, we show a further improvement of the integral Heinz mean inequality and prove \begin{eqnarray*} \frac{1}{2} \UIN{A^{2}X+2AXB+XB^{2}}\leq \frac{2}{t+2}\UIN{A^{2}X+tAXB+XB^{2}} \ \ \mbox{for all}\ \ t\in (-2, 2].\end{eqnarray*} Then we show some refinements of unitarily invariant norm inequalities, in particular we proved that: If $A, B, X \in M_{n}$ with $A$ and $B$ positive definite, and $f$, $g$ are two continuous functions on $(0,\infty)$ such that $h(x)= \frac{f(x)}{g(x)}$ is Kwong, then \begin{eqnarray*}\UIN{A^{\frac{1}{2}}(f(A)Xg(B)+g(A)Xf(B))B^{\frac{1}{2}}}\leq \frac{k}{2}\UIN{ A^{2}X+2AXB+XB^{2}}\end{eqnarray*} holds for any unitarily invariant norm, where $k=\max{\{\frac{f(\lambda)g(\lambda)}{\lambda}| \lambda \in \sigma(A)\cup \sigma(B)\}}$.