Arkiv för Matematik

  • Ark. Mat.
  • Volume 34, Number 2 (1996), 225-229.

A quantitative version of Picard's theorem

Walter Bergweiler

Full-text: Open access

Abstract

Letf be an entire function of order at least 1/2, M(r)=max|z|=r|f(z)|, and n(r, a) the number of zeros of f(z)-a in |z|≤r. It is shown that lim supr→∞n(r, a)/logM (r)≥1/2π for all except possibly one a∈C.

Note

Supported by a Heisenberg Fellowship of the Deutsche Forschungsgemeinschaft.

Article information

Source
Ark. Mat. Volume 34, Number 2 (1996), 225-229.

Dates
Received: 20 October 1995
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898512

Digital Object Identifier
doi:10.1007/BF02559545

Mathematical Reviews number (MathSciNet)
MR1416665

Zentralblatt MATH identifier
0861.30029

Rights
1996 © Institut Mittag-Leffler

Citation

Bergweiler, Walter. A quantitative version of Picard's theorem. Ark. Mat. 34 (1996), no. 2, 225--229. doi:10.1007/BF02559545. https://projecteuclid.org/euclid.afm/1485898512


Export citation

References

  • Bergweiler, W., Maximum modulus, characteristic, and area on the sphere, Analysis 10 (1990), 163–176. Erratum: Analysis 12 (1992), 67–69.
  • Borel, É., Sur les zéros des fonctions entières, Acta Math. 20 (1897), 357–396.
  • Hayman, W. K., Meromorphic Functions, Oxford Univ. Press, Oxford 1964.
  • Levin, B. J., Nullstellenverteilung ganzer Funktionen, Akademie-Verlag, Berlin, 1972.
  • Nevanlinna, R., Analytic Functions, Springer-Verlag, Berlin-Heidelberg-New York, 1970.
  • Picard, É., Sur une propriété des fonctions entières C. R. Acad. Sci. Paris 88 (1879), 1024–1027.
  • Pólya, G., Bemerkungen über unendliche Folgen und ganze Funktionen, Math. Ann. 88 (1923), 169–183.
  • Pólya, G. and Szegő, G., Aufgaben und Lehrsätze aus der Analysis, Fourth edition, Springer-Verlag, Berlin-Heidelberg-New York, 1970/71.
  • Valiron, G., Sur les fonctions entières d'ordre fini et d'ordre nul et en particulier les fonctions à correspondence régulière, Ann. Fac. Sci. Univ. Toulouse (3)5 (1913), 117–257.
  • Valiron, G., A propos d'un mémoire de M. Pólya, Bull. Sci. Math. (2)48 (1924), 9–12.