Abstract
Let P(I) be the set of all operator monotone functions defined on an interval I, and put P+(I)={h∈P(I):h(t)≥0,h≠0} and P+−1(I) = {h:h is increasing on I,h−1∈P+(0,∞)}. We will introduce a new set LP+(I)={h:h(t)>0 on I,logh∈P(I)} and show LP+(I)·P+−1(I)⊂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≤0} and if q(t) is an arbitrary factor of p(t), then p(A)2≤p(B)2 for A,B≥0 implies A2≤B2 and q(A)2≤q(B)2.
