Acta Mathematica Sinica

, Volume 12, Issue 2, pp 191–204 | Cite as

On the order of a class of meromorphic functions

  • Wu Pengcheng
Article
  • 29 Downloads

Abstract

This paper proves the following result: Letf(z) be a meromorphic function in thez-plane with a deficient value, and δ(θ k )(k=1,2, ...,q;0≤θ12<...<θq<θq+1=θ1+2π) beq rays (1≤q<∞) starting at the origin, and letn≥3 be an integer such that for any given positive numberε,0<ε<π/2,
$$\overline {\mathop {\lim }\limits_{r \to \infty } } \frac{{\log ^ + n\left\{ { \cup _{k = 1}^q \Omega \left( {\theta _k + \varepsilon ,\theta _{k + 1} - \varepsilon ,r} \right),f\prime f^n = 1} \right\}}}{{\log r}} \leqslant v< \infty ,$$
whereΝ is a constant independent ofε. IfΜ<∞, then we have
$$\lambda \leqslant \frac{\pi }{\omega } + v,$$
whereΜ andλ denote the lower order and order off(z), respectively,Ω=minθ k+1 −θ k ;1≤k≤q, andn(E, f=a) is the number of zeros off(z)−a inE with multiple zeros being counted with their multiplicities.

Keywords

Meromorphic function Order 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Edrei, A., Fuchs, W. K. J.,On meromorphic functions with region free of poles and zeros, Acta Math., 1962,108: 113–145.Google Scholar
  2. [2]
    Khuzam, F. F. A.,The order of entire functions with radially distributed zeros, Proc. Amer. Math. Soc., 1981,82: 71–75.Google Scholar
  3. [3]
    Zhang Guanghou, Wu Pengcheng,On order of meromorphic functions, Sci. Sinica, 1985,8: 785–800.Google Scholar
  4. [4]
    Hayman, W. K., Meromorphic Functions, Oxford University Press, 1964.Google Scholar
  5. [5]
    Hayman, W. K.,Angular value distribution of power series with gaps, Proc. London Math. Soc., 1972,24: 590–624.Google Scholar
  6. [6]
    Milloux, H.,SÚr une extension dÚn de P. Boutroux, H. Cartan, Bull. Soc. Math. Fr., 1937,65: 65–75.Google Scholar
  7. [7]
    Yang Lo, Values Distribution and Their New Study, Beijing: Science Press, 1982.Google Scholar
  8. [8]
    Zhang Guanghou, Entire Functions and Meromorphic Functions, Beijing: Science Press, 1986.Google Scholar
  9. [9]
    Borel, E.,Sur les zéros des fonctions entiéres, Acta Math., 1897,20: 357–396.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Wu Pengcheng
    • 1
  1. 1.Department of MathematicsGuizhou Institute for NationalitiesGuiyangChina

Personalised recommendations