An extremal property of Hermite polynomials

Abstract

Let H n be the nth Hermite polynomial, i.e., the nth orthogonal on R polynomial with respect to the weight w(x) = exp(−x 2 ). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f | |H n | at the zeros of H n+1 , then for k = 1, . . . , n we have f (k) H (k) n , where · is the L 2 (w; R) norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the L 2 (w; R) norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.

Note on an inequality of Vicente Gonçalves

“…There are a number of inequalities for the coefficients of a polynomial and its roots in the theory of polynomials (see [1][2][3][4][5][6]). The Vicente Gonçalves inequality is among the most interesting inequalities (see [4,7,8]).…”