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

TL;DR

This paper introduces new inequalities related to operator convex functions, operator means, and positive linear maps, extending classical inequalities like Davis-Choi-Jensen's and exploring their implications for operator power and geometric means.

Contribution

It provides novel complementary inequalities for operator convex functions and operator means, including bounds involving spectral radius and positive linear maps, extending existing operator inequalities.

Findings

Derived a complementary inequality to Davis-Choi-Jensen's inequality involving spectral radius.

Established bounds for operator power means under positive linear maps.

Proved inequalities for generalized geometric means with applications to operator theory.

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 $h=\max\limits_{1\leq i,j\leq n} R(A_i, A_j)$.