Skip to main content
SpringerLink
Log in

Some results on entire functions of finite lower order

  • Published:
Acta Mathematica Sinica Aims and scope Submit manuscript

Abstract

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

  1. (a)

    λ is finite;

  2. (b)

    for every arbitrary numberk 1>1, there existsk 2>1 such thatT(k 1 r,f)≤k 2 T(r,f) for allr≥r 0. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then

  3. (c)

    every deficient values off(z) is also its asymptotic value;

  4. (d)

    every asymptotic value off(z) is also its deficient value;

  5. (e)

    λ=μ;

  6. (f)

    \(\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} \)

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Drasin, D. and Weitsman, A.,Bounds on the sums of deficiencies of meromorphic functions of finite order, Complex Variables,13(1989), 111–131.

    MATH  MathSciNet  Google Scholar 

  2. Edrei, A. and Fuchs, W.H.J.,On the maximum number of deficient values of certain class of functions, Airforce Technical Report AFOSR TN, 60-402 (April 1960).

  3. Edrei, A. and Fuchs, W.H.J.,Bounds for the number of deficient values of certain classes of meromorphic functions, Proc. London Math. Soc.,12(1962), 315–344.

    Article  MATH  MathSciNet  Google Scholar 

  4. Goldberg, A.A.,Meromorphic functions with separated zeros and poles (in Russian), Izv Vyss, Ucebn. Zaved. Matemat.,17:4(1960), 67–72.

    Google Scholar 

  5. Hayman, W.K., Meromorphic Functions, Oxford, 1964.

  6. Miles, J.,On the growth of meromorphic functions with radially distributed zeros and poles, Pacific J. Math.,122(1986), 147–167.

    MATH  MathSciNet  Google Scholar 

  7. Tsuji, M., Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

    MATH  Google Scholar 

  8. Wu Shengjian, Angular Distribution and Borel Theorem of Entire and Meromorphic Functions, Dissertation, Institute of Mathematics, Academia Sinica, Beijing, 1992.

    Google Scholar 

  9. Wu Shengjian,Some problems related to the Borel directions of meromorphic functions (in Chinese), Acta Math. Sinica,6(1993), 797–804.

    Google Scholar 

  10. Yang Lo, Theory of Value-Distribution and Its New Research, (in Chinese), Science Press, Beijing, 1982.

    Google Scholar 

  11. Yang Lo,Deficient values and angular distribution of entire functions, Trans. Amer. Math. Soc.,308(1988), 583–601.

    Article  MATH  MathSciNet  Google Scholar 

  12. Zhang Guang-huo, The Theory of Entire and Meromorphic Functions-Deficient Value, Asymptotic Value and Singular Directions, (in Chinese) Science Press, Beijing 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Peking University, 100871, Beijing, China

    Wu Shengjian

Authors
  1. Wu Shengjian
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shengjian, W. Some results on entire functions of finite lower order. Acta Mathematica Sinica 10, 168–178 (1994). https://doi.org/10.1007/BF02580424

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02580424

Keywords

Search

Navigation