Derivatives of entire functions and a question of Pólya. II

Authors:
Simon Hellerstein and Jack Williamson

Journal:
Trans. Amer. Math. Soc. **234** (1977), 497-503

MSC:
Primary 30A66

MathSciNet review:
0481004

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Abstract: It is shown that if *f* is an entire function of infinite order, which is real on the real axis and has, along with , only real zeros, then has nonreal zeros (in fact, infinitely many). The finite order case was treated by the authors in a preceding paper. The combined results show that the only real entire functions *f* for which , and have only real zeros are those in the Laguerre-Pólya class, i.e.

**[1]**Simon Hellerstein and Jack Williamson,*Derivatives of entire functions and a question of Pólya*, Trans. Amer. Math. Soc.**227**(1977), 227–249. MR**0435393**, 10.1090/S0002-9947-1977-0435393-4**[2]**Simon Hellerstein and Jack Williamson,*Derivatives of entire functions and a question of Pólya*, Bull. Amer. Math. Soc.**81**(1975), 453–455. MR**0361072**, 10.1090/S0002-9904-1975-13782-7**[3]**B. Ja. Levin,*Distribution of zeros of entire functions*, American Mathematical Society, Providence, R.I., 1964. MR**0156975****[4]**B. Ja. Levin and I. V. Ostrovskiĭ,*The dependence of the growth of an entire function on the distribution of the zeros of its derivatives*, English transl., Amer. Math. Soc. Transl. (2)**32**(1963), 323-357.**[5]**R. Nevanlinna,*Le théorème de Picard-Borel et la théorie des fonctions méromorphes*, Gauthier-Villars, Paris, 1929.

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DOI:
https://doi.org/10.1090/S0002-9947-1977-0481004-1

Article copyright:
© Copyright 1977
American Mathematical Society