A new majorization between functions, polynomials, and operator inequalities

Let P+ be the set of all non-negative operator monotone functions defined on [0,∞), and put P−1 + = {h : h−1 ∈ P+}. Then P+ · P−1 + ⊂ P−1 + and P−1 + · P−1 + ⊂ P−1 + . For a function ˜h(t) and a strictly increasing function h we write ˜h h if ˜h ◦ h−1 is operator monotone. If 0 ˜h h and 0 ˜g g and if h ∈ P−1 + and g ∈ P−1 + ∪ P+, then ˜h ˜g h g. We will apply this result to polynomials and operator inequalities. Let {ai }n i=1 and {bi }n i=1 be non-increasing sequences, and put u+(t)= n i=1(t −ai ) for t a1 and v+(t)= m j=1(t −bj ) for t b1. Then v+ u+ if m n and k i=1bi k i=1ai (1 k m): in particular, for a sequence {pn}∞n =0 of orthonormal polynomials, (pn−1)+ (pn)+. Suppose 0