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January, 2008 A new majorization between functions, polynomials, and operator inequalities II
Mitsuru UCHIYAMA
J. Math. Soc. Japan 60(1): 291-310 (January, 2008). DOI: 10.2969/jmsj/06010291

## Abstract

Let $\mathbf{P}(I)$ be the set of all operator monotone functions defined on an interval $I$, and put $\mathbf{P}_{+}(I)=\{h\in \mathbf{P}(I): h(t)\geqq 0, h\neq 0\}$ and $\mathbf{P}_{+}^{-1}(I) = \{h: h$ is increasing on $I, h^{-1}\in \mathbf{P}_{+}(0,\infty)\}$. We will introduce a new set $\mathbf{L}\mathbf{P}_{+}(I)=\{h:h(t)>0$ on $I, \log h \in \mathbf{P}(I)\}$ and show $\mathbf{L}\mathbf{P}_{+}(I)\cdot \mathbf{P}_{+}^{-1}(I)\subset \mathbf{P}_{+}^{-1}(I)$ for every right open interval $I$. By making use of this result, we will establish an operator inequality that generalizes simultaneously two well known operator inequalities. We will also show that if $p(t)$ is a real polynomial with a positive leading coefficient such that $p(0)=0$ and the other zeros of $p$ are all in $\{z:Rz\leqq 0\}$ and if $q(t)$ is an arbitrary factor of $p(t)$, then $p(A)^{2}\leqq p(B)^{2}$ for $A, B\geqq 0$ implies $A^{2}\leqq B^{2}$ and $q(A)^{2}\leqq q(B)^{2}$.

## Citation

Mitsuru UCHIYAMA. "A new majorization between functions, polynomials, and operator inequalities II." J. Math. Soc. Japan 60 (1) 291 - 310, January, 2008. https://doi.org/10.2969/jmsj/06010291

## Information

Published: January, 2008
First available in Project Euclid: 24 March 2008

zbMATH: 1153.47013
MathSciNet: MR2392012
Digital Object Identifier: 10.2969/jmsj/06010291

Subjects:
Primary: 47A63
Secondary: 15A39

Keywords: Löwner-Heinz inequality , majorization , matrix order , ‎operator inequality , operator monotone function  