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 $n$, $k$.References
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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