New inequalities for operator monotone functions

Abstract

In this paper we prove that, if f:[0,∞)→R is operator monotone on [0,∞), then for all A, B such that

0<α≤A≤β<γ≤B≤δ

for some positive constants α, β, γ, δ,

0≤(γ-β)((f(δ)-f(β))/(δ-β))≤f(B)-f(A)≤(δ-α)((f(γ)-f(α))/(γ-α)).

In particular, we have the refinement and reverse of the celebrated Löwner-Heinz inequality

0<(γ-β)((δ^{r}-β^{r})/(δ-β))≤B^{r}-A^{r}≤(δ-α)((γ^{r}-α^{r})/(γ-α))

for all r∈(0,1].