Some operator mean inequalities for sector matrices

Abstract

<abstract><p>In this article, we obtain some operator mean inequalities of sectorial matrices involving operator monotone functions. Among other results, it is shown that if $ A, B\in\mathbb{M}_n(\mathbb{C}) $ are such that $ W(A), W(B)\subseteq S_{\alpha} $, $ f, g, h\in\mathfrak{m} $ are such that $ g^{\prime}(1) = h^{\prime}(1) = t $ for some $ t\in(0, 1) $ and $ 0 < mI_n\le \Re A, \Re B\le MI_n $, then</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \Re(\Phi(f(A))\sigma_h\Phi(f(B)))\le\sec^4(\alpha)K\Re \Phi(f(A\sigma_gB)), \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ M, m $ are scalars and $ \mathfrak{m} $ is the collection of all operator monotone function $ \varphi:(0, \infty)\rightarrow (0, \infty) $ satisfying $ \varphi(1) = 1 $. Moreover, we refine a norm inequality of sectorial matrices involving positive linear maps, which is a result of Bedrani, Kittaneh and Sababheh.</p></abstract>