The Normality of Meromorphic Functions with Multiple Zeros and Poles Concerning Sharing Values

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 3,  2015, pp.641-652
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.3.641
Title & Authors
The Normality of Meromorphic Functions with Multiple Zeros and Poles Concerning Sharing Values
WANG, YOU-MING;

Abstract
In this paper we study the problem of normal families of meromorphic functions concerning shared values. Let F be a family of meromorphic functions in the plane domain $\small{D{\subseteq}{\mathbb{C}}}$ and n, k be two positive integers such that $\small{n{\geq}k+1}$, and let a, b be two finite complex constants such that $\small{a{\neq}0}$. Suppose that (1) $\small{f+a(f^{(k)})^n}$ and $\small{g+a(g^{(k)})^n}$ share b in D for every pair of functions f, $\small{g{\in}F}$; (2) All zeros of f have multiplicity at least k + 2 and all poles of f have multiplicity at least 2 for each $\small{f{\in}F}$ in D; (3) Zeros of $\small{f^{(k)}(z)}$ are not the b points of f(z) for each $\small{f{\in}F}$ in D. Then F is normal in D. And some examples are provided to show the result is sharp.
Keywords
meromorphic functions;shared value;normal family;
Language
English
Cited by
References
1.
W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Revista Matemtica Iberoamericana, 11(2)(1995), 355-375.

2.
H. H. Chen and Y. X. Gu, Y, An improvement of Marty criterion and its application, Sci China Ser A, 36(1993), 674-681.

3.
J. Cline and Y. K. Hayman, The spherical derivatives of integral and meromorphic functins, Comment Math Hev, 40(1996), 117-148.

4.
G. Datt and S. Kumar, Normality of meromorphic functions with multiple zeros and poles, arXiv:1305.6214V1.

5.
H. K. Hayman, Meromorphic functions, Clarendon, Oxford, 1964.

6.
X. C. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc, 32(2000), 325-331.

7.
D. W. Meng and P. C. Hu, Normality criteria of meromorphic functons sharing one value, J. Math. Anal. Appl., 381(2011), 724-731.

8.
J. Schiff, Normal families, Springer-Verlag, Berlin, 1993.

9.
W. Schwick, Sharing Values and Normality, Archiv der Mathematik, 59(1992),50-54.

10.
Y. Xu, F. Q. Wu and L. W. Liao, Picard values and normal families of meromorphic functions, Proc. Roy. Soc. Edinburgh Sect A, 139(2009), 1091-1099.

11.
C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Science Press/Kluwer Academic, Beijing/New York, 2003.

12.
S. Zeng and I. Lahiri, A normality criterion for meromorphic functions, Kodai Math. J., 35(2012), 105-114.