## Hokkaido Mathematical Journal

### Estimates of operator convex and operator monotone functions on bounded intervals

#### 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.

#### Article information

Source
Hokkaido Math. J., Volume 45, Number 3 (2016), 325-336.

Dates
First available in Project Euclid: 7 November 2016

https://projecteuclid.org/euclid.hokmj/1478487613

Digital Object Identifier
doi:10.14492/hokmj/1478487613

Mathematical Reviews number (MathSciNet)
MR3568631

Zentralblatt MATH identifier
1372.47027

#### Citation

NAJAFI, Hamed; MOSLEHIAN, Mohammad Sal; FUJII, Masatoshi; NAKAMOTO, Ritsuo. Estimates of operator convex and operator monotone functions on bounded intervals. Hokkaido Math. J. 45 (2016), no. 3, 325--336. doi:10.14492/hokmj/1478487613. https://projecteuclid.org/euclid.hokmj/1478487613