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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The Gauss-Lucas theorem and Jensen polynomials
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by Thomas Craven and George Csordas PDF
Trans. Amer. Math. Soc. 278 (1983), 415-429 Request permission

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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Additional 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