Inverse functions of polynomials and orthogonal polynomials as operator monotone functions
Mitsuru Uchiyama · Transactions of the American Mathematical Society · 2003
We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let { p n } n = 0 ∞ \{p_n\}_{n=0}^{\infty } be a sequence of orthonormal polynomials and p n + p_{n+} the restriction of p n p_n to [ a n , ∞ ) [a_n, \infty ) , where a n a_n is the maximum zero of p n p_n . Then p n + − 1 p_{n+}^{-1} and the composite p n − 1 ∘ p n + − 1 p_{n-1}\circ p_{n+}^{-1} are operator monotone on [ 0 , ∞ ) [0, \infty ) . Furthermore, for every polynomial p p with a positive leading coefficient there is a real number