Successive derivatives of entire functions
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- by Simon Hellerstein and Jack Williamson PDF
- Proc. Amer. Math. Soc. 66 (1977), 105-108 Request permission
Abstract:
We show that if f is a real entire function which has, along with each of its derivatives, only real nonpositive zeros, then either $f(z) = c{e^{\sigma z}},c$ and $\sigma$ real constants, or \[ f(z) = c{z^m}{e^{\sigma z}}\prod \limits _n {\left ( {1 + \frac {z}{{|{z_n}|}}} \right )} \] where $\sigma \geqslant 0$ and $\sum \nolimits _n {|{z_n}{|^{ - 1}} < \infty }$.References
- Simon Hellerstein and Jack Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227–249. MR 435393, DOI 10.1090/S0002-9947-1977-0435393-4
- Simon Hellerstein and Jack Williamson, Derivatives of entire functions and a question of Pólya. II, Trans. Amer. Math. Soc. 234 (1977), no. 2, 497–503. MR 481004, DOI 10.1090/S0002-9947-1977-0481004-1 E. Laguerre, Sur les fonctions du genre zéro et du genre un, C. R. Acad. Sci. 98 (1882); Oeuvres 1 (1898), 174-177. G. Pólya, Über Annäherung durch Polynome mit reellen Wurzeln, Rend. Circ. Mat. Palermo 36 (1913), 279-295. —, Sur une question concernant les fonctions entières, C. R. Acad. Sci. Paris 158 (1914), 330-333. —, Bemerkung zur Theorie der ganzen Funktionen, Jber. Deutsch. Math. Verein. 24 (1915), 392-400.
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 66 (1977), 105-108
- MSC: Primary 30A66
- DOI: https://doi.org/10.1090/S0002-9939-1977-0460637-8
- MathSciNet review: 0460637