Complementary inequalities to Davis-Choi-Jensen's inequality and operator power means

Abstract: Let $f$ be an operator convex function on $(0,\infty)$, and $\Phi$ be a unital positive linear maps on $B(H)$. we give a complementary inequality to Davis-Choi-Jensen's inequality as follows \begin{equation*} f(\Phi(A))\geq \frac{4R(A,B)}{(1+R(A,B))^2}\Phi(f(A)), \end{equation*} where $R(A, B)=\max{r(A^{-1}B) ,r(B^{-1}A)}$ and $r(A)$ is the spectral radius of $A$. We investigate the complementary inequalities related to the operator power means and the Karcher means via unital positive linear maps, and obtain the following result: If $A_{1}$, $A_{2}$,\dots, $A_{n}$, are positive definite operators in $B(H)$, and $0<m_i\leq A_i\leq M_i$, then \begin{equation*} \Lambda( \omega;\Phi(\mathbb{A}))\geq\Phi(\Lambda( \omega; \mathbb{A}))\geq \frac{4\hbar}{(1+\hbar)^2}~\Lambda( \omega;\Phi(\mathbb{A})), \end{equation*} where $\hbar= \max\limits_{1\leq i\leq n} \frac{M_i}{m_i}$. Finally, we prove that if $G(A_1,\dots,A_n)$ is the generalized geometric mean defined by Ando-Li-Mathias for $n$ positive definite operators, then \begin{align*} \Phi(G(A_1,\dots,A_n))\geq\left(\frac{2h^\frac{1}{2}}{1+h}\right)^{n-1}G(\Phi(A_1),\dots,\Phi(A_n)), \end{align*} where $h=\max\limits_{1\leq i,j\leq n} R(A_i, A_j)$.