Normal families of meromorphic functions with multiple poles

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Published: Dec 25, 2014
DOI: https://doi.org/10.5556/j.tkjm.45.2014.1376
Keywords:
Meromorphic functions Normal family Multiplicity.

Main Article Content

Yuntong Li

Abstract

Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain $\mathcal{D}$, and $a,\ b$ be two constants such that $a\neq 0,\ \infty$ and $b\neq \infty$. If for each $f\in \mathcal{F}$, all poles of $f(z)$ are of multiplicity at least $3$ in $\mathcal{D}$, and $f'(z)+af^2(z)-b$ has at most 1 zero in $\mathcal{D}$, ignoring multiplicity, then $\mathcal{F}$ is normal in $\mathcal{D}$.

Article Details

How to Cite
Li, Y. (2014). Normal families of meromorphic functions with multiple poles. Tamkang Journal of Mathematics, 45(4), 357–366. https://doi.org/10.5556/j.tkjm.45.2014.1376
Issue
Vol. 45 No. 4 (2014)
Section
Papers
Author Biography

Yuntong Li

Department of Basic Courses, Shaanxi Railway Institute,Weinan 714000, Shaanxi Province, PR China.

References

J. Schiff, Normal Families, Springer-Verlag, Berlin,1993.

C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 1995 (In Chinese); Kluwer Academic Publishers, The Netherlands, 2003.

W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math., 70(1959), 9--42.

W. K. Hayman, Research Problems in Function Theory, Athlone Press, London, 1967.

S. Li, On normality criterion of a class of the functions, J. Fujian Normal Univ., 2 (1984), 156--158 (in Chinese).

X. Li, Proof of Hayman's conjecture on normal families, Sci. China Ser. A, 28(1985), 596--603.

J. Langley, On normal families and a result of Drasin, Proc. Roy. Soc. Edinburgh Sect. A, 98 (1984), 385--393.

X. Pang, On normal criterion of meromorphic functions, Sci. China Ser. A, 33(1990), 521--527.

H. Chen and M. Fang, On the value distribution of $f^nf'$, Sci. China Ser. A, 38(1995), 789--798.

L. Zalcman, On some questions of Hayman, unpublished manuscript, 1994.

D. Drasin, Normal families and Nevanlinna theory, Acta Math., 122 (1969), 231--263.

Y. Ye, A new criterion and its application, Chinese Ann. Math. Ser. A, 12(suppl.) (1991), 44--49 (in Chinese).

M. L. Fang and W. J. Yuan, On the normality for families of meromorphic functions,Indian J. Math., 43(2001), 341--350.

Q. Zhang, Normal families of meromorphic functions concerning shared values, J. Math. Anal. Appl., 338(2008), 545--551.

L. Zalcman, Normal families: New perspectives, Bull. Amer. Math. Soc., 35 (1998), 215--230.

W. J. Yuan and M. L. Fang, Further results of the normality for meromorphic function families, J. Guangzhou University (Natural Science Edition), 1(2002),1--9.

Y. F. Wang and M. L. Fang, Picard values and normal families of meromorphic functions with multiple zeros, Acta Math. N. S., 14(1998), 17--26. .

N. Steinmetz, Rational iteration, De Gruyter Studies in Mathematics 16, Walter de Gruyter, Berlin, New Youk, 1993.

J. M. Chang, Normality of meromorphic functions whose derivatives have $1$-points, Arch. Math., 94(2010), 555--564.