On refinements of some opeartor inequalities for positive linear maps

Abstract

We obtain some generalized inequalities in this paper. The conclusion is presented as follows:
Let $0<mI\le A\le m^{\prime}I\le M^{\prime}I\le B\le MI$ and $p\ge1$. Then for every positive unital linear map $\Phi$,\\
\begin{center} $\begin{array}{rlr}
\Phi^{2p}(A\nabla_t B)\le(\frac{K(h,2)}{4^{\frac{1}{p}-1}(1+Q(t)(\log \frac{M^{\prime}}{m^{\prime}})^2)})^{2p}\Phi^{2p}(A\sharp_t B)\end{array}$
\end{center}
and
\begin{center} $\begin{array}{rlr}
\Phi^{2p}(A\nabla_t B)\le(\frac{K(h,2)}{4^{\frac{1}{p}-1}(1+Q(t)(\log \frac{M^{\prime}}{m^{\prime}})^2)})^{2p}(\Phi(A)\sharp_t \Phi(B))^{2p},\end{array}$
\end{center}
where $t\in[0,1]$, $h=\frac{M}{m}$, $Q(t)=\frac{t^2}{2}(\frac{1-t}{t})^{2t}$ and $Q(0)=Q(1)=0$. Moreover, we give an improvement for the operator version of Wielandt inequality.