Estimates of operator convex and operator monotone functions on bounded intervals

TL;DR

This paper investigates the behavior of operator convex and monotone functions on bounded intervals, providing new inequalities, bounds, and extensions related to these functions in the context of bounded linear operators.

Contribution

It offers new results on inequalities involving operator convex and monotone functions on bounded intervals, including a lower bound for differences and an extension of the Furuta inequality.

Findings

Proved a strict inequality for nonlinear operator convex functions on bounded intervals.

Derived a lower bound for the difference of operator monotone functions.

Provided an estimation of the Furuta inequality extending the L"owner--Heinz inequality.

Abstract

Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,\infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the L\"owner--Heinz inequality.