Inverse Functions of Polynomials and Orthogonal Polynomials as Operator Monotone Functions

We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let {pn}n=0 be a sequence of orthonormal polynomials and pn+ the restriction of pn to [an,∞), where an is the maximum zero of pn. Then p −1 n+ and the composite pn−1 ◦ p −1 n+ are operator monotone on [0,∞). Furthermore, for every polynomial p with a positive leading coefficient there is a real number a so that the inverse function of p(t+a)−p(a) defined on [0,∞) is semi-operator monotone, that is, for matrices A,B ≥ 0, (p(A+ a)− p(a))2 ≤ ((p(B + a)− p(a))2 implies A2 ≤ B2.