Some operator mean inequalities for sector matrices

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 $g^{\mathrm{\prime }}(1)=h^{\mathrm{\prime }}(1)=t$ for some $t\in (0,1)$ and $0<m{I}_n\le \mathrm{\Re }A,\mathrm{\Re }B\le M{I}_n$, then

$$\begin{array}{r}\mathrm{\Re }(\mathrm{\Phi }(f(A)){\sigma }_h\mathrm{\Phi }(f(B)))\le {\sec }^4(\alpha )K\mathrm{\Re }\mathrm{\Phi }(f(A{\sigma }_{g}B)),\end{array}$$

where $M,m$ are scalars and $\mathfrak{m}$ is the collection of all operator monotone function $\phi :(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ satisfying $\phi (1)=1$. Moreover, we refine a norm inequality of sectorial matrices involving positive linear maps, which is a result of Bedrani, Kittaneh and Sababheh.