Let be the set of all operator monotone functions defined on an interval , and put and is increasing on . We will introduce a new set on and show for every right open interval . 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 is a real polynomial with a positive leading coefficient such that and the other zeros of are all in and if is an arbitrary factor of , then for implies and .
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