Peter C. Gibson,Dept. of Mathematics & Statistics, York University, 4700 Keele St., Toronto, Ontario,
Canada, M3J 1P3; e-mail: pcgibson@yorku.ca

Abstract/Résumé:

The Pólya-Schur problem for a region \(Z\) in the complex plane is to characterize the semigroup of linear operators \(A:\mathbb{C}[z]\rightarrow \mathbb{C}[z]\) that map polynomials whose zeros are confined to \(Z\) to polynomials of the same type, or to 0. We give a constructive solution to the Pólya-Schur problem in the case where \(Z\) is the unit circle. This shows that the associated semigroup is qualitatively simpler than in the classical case where \(Z\) is the real line, whereas recent results have not clearly distinguished the two cases.

Le problème Pólya-Schur pour une région \(Z\) dans le plan complexe est de charactériser le semigroupe des opérateurs linéaires \(A:\mathbb{C}[z]\rightarrow \mathbb{C}[z]\) envoyant chaque polynôme dont les racines appartiennent à \(Z\) vers un polynôme du même type, ou vers 0. Nous présentons une solution constructive au problème Pólya-Schur dans le cas où \(Z\) est le cercle unité. Cela démontre que le semigroupe associé est qualitativement plus simple que dans le cas classique de la ligne réelle, tandis que les résultats récents n’ont pas distingué les deux cas.

Keywords: Polya-Schur type theorems, composition operators, stable polynomials

AMS Subject Classification: Polynomials, Zeros of polynomials; rational functions; and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral), Operators on function spaces (general) 30C10, 30C15, 47B38

PDF(click to download): The Pólya-Schur Problem on the Unit Circle

Faruk F. Abi-Khuzam, Department of Mathematics, American University of Beirut, P.O. Box 11-0236, Riad El Solh, Beirut 1107 2020, Lebanon; email: farukakh@aub.edu.lb

May F. Hamdan, Division of Computer Science and Mathematics, Lebanese American University, P.O. Box 135053 F 64, Beirut, Lebanon; email: mhamdan@lau.edu.lb

Abstract/Résumé:

For an entire function \(f\) with non-negative Maclaurin coefficients, a region is obtained which is defined in terms of Hayman’s function \(b(r) = r (rf^{\prime} (r)/f(r))^{\prime}\), and which is free of all zeros of \(f\) and those of all its sections. The new region defined improves on previous results. In particular, it is shown that when \(\underset{n\rightarrow \infty}{\limsup}\, b(r) = A^2/4\), \(A>0\), then the zeros \(r_n \exp (i\theta_n)\) of \(f\) satisfy the inequality, \(\underset{n\rightarrow \infty}{\liminf}\, |\theta_n| \geq 4\sin^{-1} (1/A\sqrt{2})\), which is very close to being optimal.

Etant donnée une function entière \(f\) avec des coéfficients positifs, on trouve une région définie en termes de la fonction \(b(r) = r (rf^{\prime}(r)/f(r))^{\prime}\) de Hayman, dépourvue des zéros de \(f\) et de ceux de toutes ses sections. Particulièrement, on démontre qu’au cas où \(\underset{n\rightarrow \infty}{\limsup}\, b(r) = A^2/4\), \(A>0\), les zéros \(r_n \exp (i\theta_n)\) de \(f\) satisfont l’inégalité \(\underset{n\rightarrow\infty}{\liminf} \, |\theta_n| \geq 4\sin^{-1} (1/A\sqrt{2})\), qui est presque optimale.

Keywords:

AMS Subject Classification: Zeros of polynomials; rational functions; and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) 30C15

PDF(click to download): Radial Distribution of Zeros of Entire Functions and Sections of their Power Series

P. Lakatos

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords: Chebyshev transform, reciprocal, semi-reciprocal polynomials, zeros on the unit circle

AMS Subject Classification: Zeros of polynomials; rational functions; and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) 30C15

PDF(click to download): On polynomials having zeros on the unit circle