Further Generalizations of Some Operator Inequalities Involving Positive Linear Map

Abstract

We obtain a generalized conclusion based on an $\alpha$-geometric mean inequality. The conclusion is presented as follows: If $m_1, M_1, m_2, M_2$ are positive real numbers, $0<m_1\le A\le M_1$ and $0<m_2\le B\le M_2$ for $m_1<M_1$ and $m_2<M_2$, then for every unital positive linear map $\Phi$ and $\alpha\in (0,1]$, the operator inequality below holds:\\
 $(\Phi(A)\sharp_\alpha\Phi(B))^p\le \frac{1}{16}\left\{ \frac{(M_1+m_1)^2((M_1+m_1)^{-1}(M_2+m_2))^{2\alpha})}{(m_2M_2)^\alpha(m_1M_1)^{1-\alpha}} \right\}^p\Phi^p(A\sharp_\alpha B),\quad p\ge2.$\\
 Likewise, we give a second powering of the Diaz-Metcalf type inequality. Finally, we present $p-$th powering of some reversed inequalities for $n$ operators related to Karcher mean and power mean involving positive linear maps.