Some bounds for means of accretive matrices

In this paper, by using the bounds of the real part of sector matrices A and B, and |A|, |B|, we obtain some bounds for the real part of \(A\sigma B\) $A\sigma B$ . Among them, let \(A,B\in {\mathcal S_{\alpha }}\) $A,B\in {\mathcal{S}}_{\alpha }$ such that \(0<mI\le |A|,|B|\le MI\) $0<mI\le |A|,|B|\le MI$ . If \(p\notin [0,2)\) $p\notin [0,2)$ , then for every two matrix means \(\sigma _1\le \sigma _2\) ${\sigma }_1\le {\sigma }_2$ , we have 1.1 \(\begin{aligned} \Re (A\sigma _1 B)\le K_p^{\frac{1}{p}}\sec ^2\alpha \left( |A|\sigma _2 |B|\right) , \end{aligned}\) $$\begin{array}{c}\hfill \Re (A{\sigma }_{1}B)\le K_p^{\frac{1}{p}}{sec}^2\alpha \left(|A|,{\sigma }_2,|B|\right),\end{array}$$ where \(K_p=K(m,M,p)=\frac{(mM^p-Mm^p)}{(p-1)(M-m)}\left( \frac{p-1}{p}\frac{M^p-m^p}{mM^p-Mm^p}\right) ^p\) $K_p=K(m,M,p)=\frac{(m{M}^p-M{m}^p)}{(p-1)(M-m)}{\left(\frac{p-1}{p},\frac{M^p-m^p}{m{M}^p-M{m}^p}\right)}^p$ .