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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The Gauss-Lucas theorem and Jensen polynomials
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by Thomas Craven and George Csordas
Trans. Amer. Math. Soc. 278 (1983), 415-429
DOI: https://doi.org/10.1090/S0002-9947-1983-0697085-9

Abstract:

A characterization is given of the sequences $\{ {\gamma _k}\}_{k = 0}^\infty$ with the property that, for any complex polynomial $f(z) = \Sigma {a_k}{z^k}$ and convex region $K$ containing the origin and the zeros of $f$, the zeros of $\Sigma {\gamma _k}{a_k}{z^k}$ again lie in $K$. Many applications and related results are also given. This work leads to a study of the Taylor coefficients of entire functions of type $\text {I}$ in the Laguerre-Pólya class. If the power series of such a function is given by $\Sigma {\gamma _k}{z^k}/k!$ and the sequence $\{ {\gamma _k}\}$ is positive and increasing, then the sequence satisfies an infinite collection of strong conditions on the differences, namely ${\Delta ^n}{\gamma _k} \geqslant 0$ for all $n$, $k$.
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Bibliographic Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 278 (1983), 415-429
  • MSC: Primary 30D10; Secondary 12D05, 30C15
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0697085-9
  • MathSciNet review: 697085