A new majorization between functions, polynomials, and operator inequalities II

Let P(I) be the set of all operator monotone functions defined on an interval I, and put P+(I) = {h is an element of P(I): h(t) >= 0, h not equal 0} and P-+(-1)(I) = {h: h is increasing on I, h(-1) is an element of P+(0, infinity)}. We will introduce a new set LP+(I)= {h: h(t) > 0 on I, log h is an element of P(I)} and show LP+(I) . P-+(-1) (I) subset of 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 A(2) <= B-2 and q(A)(2) <= q(B)(2).