Hokkaido Mathematical Journal

Estimates of operator convex and operator monotone functions on bounded intervals

Hamed NAJAFI, Mohammad Sal MOSLEHIAN, Masatoshi FUJII, and Ritsuo NAKAMOTO

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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

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

First available in Project Euclid: 7 November 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A63: Operator inequalities
Secondary: 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] 47A30: Norms (inequalities, more than one norm, etc.)

L\"owner--Heinz inequality, Furuta inequality and operator monotone function Furuta inequality and operator monotone function


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

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