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Transactions of the American Mathematical Society

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The Gauss-Lucas theorem and Jensen polynomials

Authors: Thomas Craven and George Csordas
Journal: Trans. Amer. Math. Soc. 278 (1983), 415-429
MSC: Primary 30D10; Secondary 12D05, 30C15
MathSciNet review: 697085
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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 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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Keywords: Multiplier sequences, Jensen polynomials, entire functions, composite polynomials, roots of polynomials, differential operators
Article copyright: © Copyright 1983 American Mathematical Society

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