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2014 A characterization of convex functions and its application to operator monotone functions
Masatoshi Fujii, Young Ok Kim, Ritsuo Nakamoto
Banach J. Math. Anal. 8(2): 118-123 (2014). DOI: 10.15352/bjma/1396640056

Abstract

We give a characterization of convex functions in terms of difference among values of a function. As an application, we propose an estimation of operator monotone functions: If $A \geq B \ge 0$, $A-B$ is invertible and $f$ is operator monotone on $(0, \infty)$, then $ f(A) - f(B) \ge f(\|B\|+ \epsilon) - f(\|B\|) > 0$, where $\epsilon = \|(A-B)^{-1}\|^{-1}$. Moreover it gives a simple proof to Furuta's theorem: If $\log A$ is strictly greater than $\log B$ for invertibel operators $A, \ B \geq 0$ and $f$ is operator monotone on $(0, \infty)$, then there exists a positive number $\beta$ such that $ f(A^\alpha)$ is strictly greater than $f(B^\alpha)$ for all positive numbers $ \alpha \le \beta$.

Citation

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Masatoshi Fujii. Young Ok Kim. Ritsuo Nakamoto. "A characterization of convex functions and its application to operator monotone functions." Banach J. Math. Anal. 8 (2) 118 - 123, 2014. https://doi.org/10.15352/bjma/1396640056

Information

Published: 2014
First available in Project Euclid: 4 April 2014

zbMATH: 1304.47024
MathSciNet: MR3189543
Digital Object Identifier: 10.15352/bjma/1396640056

Subjects:
Primary: 47A63
Secondary: 47B10, 47BA30

Rights: Copyright © 2014 Tusi Mathematical Research Group

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Vol.8 • No. 2 • 2014
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