Derivatives of entire functions and a question of Pólya
Authors:
Simon Hellerstein and Jack Williamson
Journal:
Trans. Amer. Math. Soc. 227 (1977), 227-249
MSC:
Primary 30A64
DOI:
https://doi.org/10.1090/S0002-9947-1977-0435393-4
MathSciNet review:
0435393
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Abstract | References | Similar Articles | Additional Information
Abstract: An old question of Pólya asks whether an entire function f which has, along with each of its derivatives, only real zeros must be of the form \[ f(z) = {z^m}{e^{ - a{z^2} + bz + c}}\prod \limits _n {\left ( {1 - \frac {z}{{{z_n}}}} \right )} {e^{z/{z_n}}}\] where $a \geqslant 0,b$ and the ${z_n}$ are real, and ${\Sigma _n}z_n^{ - 2} < \infty$. This note answers this question (essentially in the affirmative) if f is of finite order; indeed, it is established that if $f,f’$, and $f''$ have only real zeros (f of finite order), then either f has the above form or f has one of the forms \[ f(z) = a{e^{bz}},\quad f(z) = a({e^{icz}} - {e^{id}})\] where a, b, c, and d are constants, b complex, c and d real.
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Additional Information
Keywords:
Real entire function,
finite order,
finite genus,
derivative
Article copyright:
© Copyright 1977
American Mathematical Society