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 $\{p_n\}_{n=0}^{\infty}$ be a sequence of orthonormal polynomials and $p_{n+}$ the restriction of $p_n$ to $[a_n, \infty)$, where $a_n$ is the maximum zero of $p_n$. Then $p_{n+}^{-1}$ and the composite $p_{n-1}\circ p_{n+}^{-1}$ are operator monotone on $[0, \infty)$. 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 $[0,\infty)$is semi-operator monotone, that is, for matrices $A,B \geq 0$, $(p(A+a)-p(a))^2 \leq ((p(B+a)-p(a))^{2}$ implies $A^2\leq B^2.$