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

DOI10.1007/S40315-015-0151-XzbMATH Open1359.30012arXiv1405.3682OpenAlexW1492447913MaRDI QIDQ312718

Martin Lamprecht

Publication date: 16 September 2016

Published in: (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1405.3682

zbMATH Keywords

zeros of polynomials\(*_\lambda\)-convolution

Mathematics Subject Classification ID

Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) (30C15) Polynomials and rational functions of one complex variable (30C10)

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