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

The Sz\'asz inequality is a classical result providing a bound for polynomials with zeros in the upper half of the complex plane in terms of its low-order coefficients. Some generalisations of this result to polynomials in several variables were done by Borcea-Br\"and\'en and Knese. In this article the inequality is discussed in the context of matrix polynomials and matrix variables. As a byproduct we improve the scalar Sz\'asz inequality by relaxing the assumption of the location of zeros.