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 {p(n)}(n=0)(infinity) be a sequence of orthonormal polynomials and p(n+) the restriction of p(n) to [a(n); infinity), where a(n) is the maximum zero of pn. Then p(n+)(-1) and the composite p(n-1) circle p(n+)(-1) are operator monotone on [ 0; 1). 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; infinity) is semi-operator monotone, that is, for matrices A; B greater than or equal to 0, (p(A + a) -p(a))(2) less than or equal to ((p(B + a)-p(a))(2) implies A(2) less than or equal to B-2.