Suffridge's convolution theorem for polynomials with zeros in the unit disk

TL;DR

This paper explores extensions of Suffridge's convolution theorem for polynomials with zeros in the unit disk, aiming to generalize it further and identify invariant zero domains under the Grace-Szeg"o convolution.

Contribution

The authors extend Suffridge's theorem to certain classes of polynomials with zeros in the unit disk and identify new zero domains invariant under the Grace-Szeg"o convolution.

Findings

Suffridge's convolution theorem holds for specific polynomial classes with zeros in the unit disk.

Identification of non-circular zero domains invariant under the Grace-Szeg"o convolution.

Partial extension of the Grace-Szeg"o convolution theorem.

Abstract

In 1976 Suffridge proved an intruiging theorem regarding the convolution of polynomials with zeros only on the unit circle. His result generalizes a special case of the fundamental Grace-Szeg\"o convolution theorem, but so far it is an open problem whether there is a Suffridge-like extension of the general Grace-Szeg\"o convolution theorem. In this paper we try to approach this question from two different directions: First, we show that Suffridge's convolution theorem holds for a certain class of polynomials with zeros in the unit disk and thus obtain an extension of one further special case of the Grace-Szeg\"o convolution theorem. Second, we present non-circular zero domains which stay invariant under the Grace-Szeg\"o convolution hoping that this will lead to further analogs of Suffridge's convolution theorem.