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(Φ(A))\geq \frac{4R(A,B)}{(1+R(A,B))^2}Φ(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 $0<m_i\le A_i\le M_i$, then \begin{equation*} Λ( ω;Φ(\mathbb{A}))\geqΦ(Λ( ω; \mathbb{A}))\geq \frac{4\hbar}{(1+\hbar)^2}~Λ( ω;Φ(\mathbb{A})), \end{equation*} where $\hslash =\underset{1\le i\le n}{\max }\frac{M_i}{m_i}$. 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*} Φ(G(A_1,\dots,A_n))\geq\left(\frac{2h^\frac{1}{2}}{1+h}\right)^{n-1}G(Φ(A_1),\dots,Φ(A_n)), \end{align*} where $h=\underset{1\le i,j\le n}{\max }R(A_i,A_j)$.