The Sz\'asz inequality for matrix polynomials and functional calculus

Abstract

The Sz\'asz inequality is a classical result that provides a bound for polynomials with zeros in the upper half of the complex plane, expressed in terms of their low-order coefficients. Generalizations of this result to polynomials in several variables have been obtained by Borcea-Br\"and\'en and Knese. In this article, we discuss the Sz\'asz inequality in the context of polynomials with matrix coefficients or matrix variables. In the latter case, the estimation provided by the Sz\'asz-type inequality can be sharper than that offered by the von Neumann inequality. As a byproduct, we improve the scalar Sz\'asz inequality by relaxing the assumption regarding the location of zeros.