SEVERAL INEQUALITIES FOR OPERATOR MONOTONE FUNCTIONS ON FINITE INTERVALS

Abstract

In this paper we show that, if the function \(f:(-1,1)\to \mathbb{R}\) is operator monotone in \((-1,1)\), then there exists a positive measure \(\mu\) on \([-1,1]\) such that
\[
[f(B)-f(A)](B-A)
= f'(0)\int_{-1}^{1}\left(\int_{0}^{1}\left[(1-\lambda((1-t)A+tB))^{-1}(B-A)\right]^2\,dt\right)d\mu(\lambda)
\]
for all \(A,B\) with \(\operatorname{Sp}(A), \operatorname{Sp}(B)\subset(-1,1)\). Some necessary and sufficient conditions for the operators \(A,B\) with \(\operatorname{Sp}(A), \operatorname{Sp}(B)\subset(-1,1)\) such that the inequality
\[
f(B)B+f(A)A\geq f(A)B+f(B)A
\]
holds for any operator monotone function \(f\) on \((-1,1)\) are also given.