Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A proof of the Pólya-Wiman conjecture


Author: Young-One Kim
Journal: Proc. Amer. Math. Soc. 109 (1990), 1045-1052
MSC: Primary 30D20; Secondary 30D35
DOI: https://doi.org/10.1090/S0002-9939-1990-1013971-3
MathSciNet review: 1013971
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f(z) = {e^{ - \alpha {z^2}}}g(z)$ where $ \alpha \geq 0$ and $ g$ is a real entire function of genus at most 1. It is shown that if $ f$ has only a finite number of nonreal zeros, then its derivatives, from a certain one onward, have only real zeros.


References [Enhancements On Off] (What's this?)

  • [B] R. P. Boas, Entire functions, Academic Press, New York, 1954. MR 0068627 (16:914f)
  • [CCS1] T. Craven, G. Csordas, and W. Smith, The zeros of derivatives of entire functions and the Pólya-Wiman Conjecture, Ann. of Math. 125 (1987), 405-431. MR 881274 (88a:30007)
  • [CCS2] -, Zeros of derivatives of entire functions, Proc. Amer. Math. Soc. 101 (1987), 323-326. MR 902550 (88k:30024)
  • [G] W. Gontcharoff, Recherches sur les dérivées successives des fonctions analytiques, Ann. École Norm. 47 (1930), 1-78. MR 1509300
  • [L] E. Laguerre, Oeuvres I, Gauthier-Villars, Paris, 1898.
  • [Le] B. Ja. Levin, Distribution of zeros of entire functions, Trans. Math. Monographs, vol. 5, Amer. Math. Soc., Providence, RI, 1964. MR 0156975 (28:217)
  • [P1] G. Pólya, Über Annäherung durch Polynome mit lauter reellen Wurzeln, Rend. Circ. Mat. Palermo 36 (1913), 279-295.
  • [P2] -, Some problems connected with Fourier's work on transcendental equations, Quart. J. Math. Oxford Ser. 1 (1930), 21-34.
  • [P3] -, Über die Realität der Nullstellen fast aller Ableitungen gewisser ganzer Funktionen, Math. Ann. 114 (1937), 622-634. MR 1513163
  • [P4] -, On the zeros of derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178-191. MR 0007781 (4:192d)
  • [Wi1] A. Wiman, Über eine asymptotische Eigenschaft der Ableitungen der ganzen Funktionen von den Geschlechtern 1 und 2 mit einer endlichen Anzahl von Nullstellen, Math. Ann. 104 (1930), 169-181. MR 1512660
  • [Wi2] -, Über die Realität der Nullstellen fast aller Ableitungen gewisser ganzer Funktionen, Math. Ann. 114 (1937), 617-621. MR 1513162

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30D20, 30D35

Retrieve articles in all journals with MSC: 30D20, 30D35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1013971-3
Keywords: Pólya-Wiman conjecture, Laguerre-Pólya class
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society