Computational Methods and Function Theory

, Volume 2, Issue 1, pp 257–265

# Normal Families of Meromorphic Functions whose Derivatives Omit a Function

• Xuecheng Pang
• Degui Yang
• Lawrence Zalcman
Article

## Abstract

Let $$\cal F$$ be a family of functions meromorphic on the plane domain D, and let h be a holomorphic function on D, h n= 0. Suppose that, for each $$f \in {\cal F}$$, f (m)(z) ≠ h(z) for zD. Then $$t\cal F$$ is normal on D (i) if all zeros of functions in $$\cal F$$ have multiplicity at least m + 3, or (ii) if all zeros of functions in $$\cal F$$ have multiplicity at least m + 2 and h has only multiple zeros on D, or (iii) if all poles of functions in $$\cal F$$ are multiple and all zeros have multiplicity at least m + 2.

## En]Keywords

Normal families omitted functions

30D45

## References

1. 1.
W. Bergweiler, On the zeros of certain homogeneous differential polynomials, Arch. Math. 64 (1995), 199–202.
2. 2.
W. Bergweiler, Normality and exceptional values of derivatives, Proc. Amer. Math. Soc. 120 (2001), 121–129.
3. 3.
W. Bergweiler and A. Eremenko, On the singularities of the inverse of a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), 355–373.
4. 4.
X. Hua, On a problem of Hayman, Kodai Math. J. 13 (1990), 386–390.
5. 5.
S. Nevo, On theorems of Yang and Schwick, Complex Variables 46 (2001), 315–321.
6. 6.
S. Nevo, Applications of Zalcman’s Lemma to Q m-normal families, Analysis 21 (2001), 289–325.
7. 7.
X. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc. 32 (2000), 325–331.
8. 8.
—, Normal families of meromorphic functions with multiple zeros and poles, to appear in Israel J. Math. Google Scholar
9. 9.
W. Schwick, Exceptional functions and normality, Bull. London Math. Soc. 29 (1997), 425–432.
10. 10.
Y. Wang and M. Fang, Picard values and normal families of meromorphic functions with multiple zeros, Acta Math. Sinica N. S. 14 (1998), 17–26.
11. 11.
L. Yang, Normality for families of meromorphic functions, Sci. Sinica Ser. A 29 (1986), 1263–1274.
12. 12.
L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. 35 (1998), 215–230.