Inequalities for sector matrices with negative power

Abstract

In this paper, we present some inequalities for sector matrices with negative power. Among other results, we prove that if $A,B\in\mathbb{M}_n(\mathbb{C})$ such that $W(A),W(B)\subseteq S_{\alpha}$, then
\begin{eqnarray*}
\cos^2(\alpha)\Re(vA^{-1}+(1-v)B^{-1})^{-1}\le\Re(vA+(1-v)B),
\end{eqnarray*}
where $v\in[0,1]$. This considerablely improves Tan and Xie's Theorem 2.4 in \cite{TX} and Bedrani, Kittaneh and Sababheh's Theorem 4.1 in \cite{BKS}.