Mathematical Notes

, Volume 75, Issue 3–4, pp 410–417 | Cite as

Asymptotic Relations between Maximums of Absolute Values and Maximums of Real Parts of Entire Functions

  • P. V. Filevich
Article

Abstract

According to the classical Wiman--Valiron theorem, the maximum of the absolute value and the maximum of the real part of an entire function are asymptotically equal at infinity outside an exceptional set of a finite logarithmic measure. In this paper, we study the following problem concerning the exceptional set: how does the ratio of the maximum of the absolute value and the maximum of the real part of an entire function depend on its Taylor coefficients? In particular, our results imply that the maximum of the absolute value can increase arbitrarily fast with respect to the maximum of the real part or the Nevanlinna characteristic.

Wiman--Valiron theorem Nevanlinna characteristic exceptional set entire function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    M. N. Sheremeta, “On relations between the maximal summand and the maximum of the absolute value of entire Dirichlet series,” Mat. Zametki [Math. Notes], 51 (1992), no. 5, 141–148.Google Scholar
  2. 2.
    G. Valiron, Lectures on the General Theory of Integral Functions, Édouard Privat, Toulouse, 1923.Google Scholar
  3. 3.
    G. Vittikh, The Latest Investigations on One-Valued Analytical Functions [in Russian], Fizmatgiz, Moscow, 1960.Google Scholar
  4. 4.
    W. Bergweiler, “On meromorphic functions that share three values and on the exceptional set in Wiman—Valiron theory,” Kodai Math. J., 13 (1990), no. 1, 1–9.Google Scholar
  5. 5.
    O. B. Skaskiv and P. V. Filevich, “On the size of the exceptional set in Wiman's theorem,” Matem. Studii [in Ukrainian], 12 (1999), no. 1, 31–36.Google Scholar
  6. 6.
    W. Hayman, Meromorphic Functions, Oxford, 1964; Russian translation: Mir, Moscow, 1966.Google Scholar
  7. 7.
    Sh. I. Strelitz, Asymptotic Properties of Analytical Solutions of Differential Equations [in Russian], Mintis, Vilnius, 1972.Google Scholar
  8. 8.
    G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, 3d ed., Springer-Verlag, Berlin—Göttingen—Heidelberg—New York, 1964. Russian translation: vol. 2, Nauka, Moscow, 1978.Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • P. V. Filevich
    • 1
  1. 1.I. Franko Lviv State UniversityRussia

Personalised recommendations