## Banach Journal of Mathematical Analysis

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

#### 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 $\alpha \le \beta$.

#### Article information

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

Dates
First available in Project Euclid: 4 April 2014

https://projecteuclid.org/euclid.bjma/1396640056

Digital Object Identifier
doi:10.15352/bjma/1396640056

Mathematical Reviews number (MathSciNet)
MR3189543

Zentralblatt MATH identifier
1304.47024

#### Citation

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. https://projecteuclid.org/euclid.bjma/1396640056

#### References

• J.I. Fujii, M. Fujii and R. Nakamoto, An operator inequality implying the usual and chaotic orders, Ann. Funct. Anal. 5 (2014), 24–25.
• M. Fujii, T. Furuta and E. Kamei, Furuta's inequality and its application to Ando's Theorem, Linear Algebra Appl. 179 (1993), 161–169.
• M. Fujii, J.-F. Jiang and E. Kamei, Characterization of chaotic order and its application to Furuta inequality, Proc. Amer. Math. Soc. 125 (1997), 3655–3658.
• M. Fujii, Furuta inequality and its related topics, Ann. Funct. Anal. 1 (2010), 28–45.
• T. Furuta, $A \ge B \ge 0$ assures $(B^rA^pB^r)^{1/q} \ge B^{(p+2r)/q}$ for $r \ge 0,\ p \ge 0,\ q \ge 1$ with $(1 + 2r)q \ge p + 2r$, Proc. Amer. Math. Soc. 101 (1987), 85–88.
• T. Furuta, Operator monotone functions, $A>B\ge 0$ and $\log A > \log B$, J. Math. Inequal.7 (2013), 93–96.
• T. Furuta, Comprehensive survey on an order preserving operator inequality, Banach J. Math. Anal. 7 (2013), 14–40.
• D.S. Mitrinović, J.E. Pečarić and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.
• M.S. Moslehian and H. Najafi, An extension of the Löwner–Heinz inequality, Linear Algebra Appl. 437 (2012), 2359–2365.
• M. Uchiyama, Strong monotonicity of operator functions, Integral Equations Operator Theory 37 (2000), 95–105.
• T. Yamazaki, The Riemannian mean and matrix inequalities related to the Ando-Hiai inequality and chaotic order, Oper. Matrices 6 (2012), 577–588.