Banach Journal of Mathematical Analysis

A characterization of convex functions and its application to operator monotone functions

Masatoshi Fujii, Young Ok Kim, and Ritsuo Nakamoto

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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$.

Article information

Banach J. Math. Anal., Volume 8, Number 2 (2014), 118-123.

First available in Project Euclid: 4 April 2014

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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] 47BA30

Convex function operator monotone function Lówner-Heinz inequality chaotic order


Fujii, Masatoshi; Kim, Young Ok; Nakamoto, Ritsuo. A characterization of convex functions and its application to operator monotone functions. Banach J. Math. Anal. 8 (2014), no. 2, 118--123. doi:10.15352/bjma/1396640056.

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