Hokkaido Mathematical Journal, ( ISI ), Volume (45), No (3), Year (2016-9) , Pages (325-336)

Title : ( Estimates of operator convex and operator monotone functions on bounded intervals )

Authors: Hamed Najafi , Mohammad Sal Moslehian , M. Fujii , Ritsuo Nakamoto ,

Citation: BibTeX | EndNote

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.

Keywords

, L\"owner--Heinz inequality, Furuta inequality and operator monotone function
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@article{paperid:1060668,
author = {Najafi, Hamed and Sal Moslehian, Mohammad and M. Fujii and Ritsuo Nakamoto},
title = {Estimates of operator convex and operator monotone functions on bounded intervals},
journal = {Hokkaido Mathematical Journal},
year = {2016},
volume = {45},
number = {3},
month = {September},
issn = {0385-4035},
pages = {325--336},
numpages = {11},
keywords = {L\"owner--Heinz inequality; Furuta inequality and operator monotone function},
}

%0 Journal Article
%T Estimates of operator convex and operator monotone functions on bounded intervals
%A Najafi, Hamed