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 . Then and . For a function and a strictly increasing function h we write if is operator monotone. If and and if and , then . We will apply this result to polynomials and operator inequalities. Let and be non-increasing sequences, and put for t≧a1 and for t≧b1. Then v+⪯u+ if m≦n and : in particular, for a sequence of orthonormal polynomials, (pn-1)+⪯(pn)+. Suppose 0