## Normal families and shared values of meromorphic functions. II.(English)Zbl 1045.30020

Let $$F$$ be a family of functions meromorphic in a domain $$D$$, let $$k$$ be a positive integer and let $$b,c,d$$ be nonzero complex numbers such that $$b\not=d$$. Suppose that for all $$f\in F$$ the zeros of $$f$$ have multiplicity at least $$k$$. Suppose also that $$f^{(k)}(z)=b$$ whenever $$f(z)=0$$ and that $$f(z)=c$$ whenever $$f^{(k)}(z)=d$$. It is shown that $$F$$ is normal if either $$k\geq 3$$, or $$k=2$$ and $$b\not= 4d$$, or $$k=1$$ and $$b\not=(m+1)d$$ for all positive integers $$m$$. This result extends various other results obtained previously by the authors and others.
Interesting examples show that the extra conditions in the case $$k=2$$ and $$k=1$$ are needed. The main tool used is a rescaling lemma introduced by L. Zalcman, later extended by X. C. Pang, and then further refined by X. C. Pang and L. Zalcman [Bull. Lond. Math. Soc. 32, 325–331 (2000; Zbl 1030.30031)].
A second result concerns the normality of families of functions $$f$$ for which $$f(z)f^{(k)}(z)=a$$ whenever $$f^{(k)}(z)=b$$, and vice versa.

### MSC:

 30D45 Normal functions of one complex variable, normal families 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Zbl 1030.30031
Full Text:

### References:

 [1] W. Bergweiler, On the zeros of certain homogeneous differential polynomials, Arch. Math. 64 (1995), 199–202. · Zbl 0818.30021 [2] W. Bergweiler Normality and exceptional values of derivatives, Proc. Amer. Math. Soc. 129 (2000), 121–129. · Zbl 0961.30025 [3] W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order Rev. Mat. Iber. 11 (1995), 355–373. · Zbl 0830.30016 [4] H. H. Chen and M. L. Fang, On the value distribution of fnf’, Sci. China Ser. A 38 (1995), 789–798. · Zbl 0839.30026 [5] M. L. Fang and L. Zalcman, Normal families and shared values of meromorphic functions to appear in Ann. Polon. Math. [6] Y. X. Gu, Un critére de normalité des familles de fonctions méromorphes, Sci. Sinica Special Issue I (1979), 267–274. [7] W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. 2 70 (1959), 9–42. · Zbl 0088.28505 [8] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [9] X. C. Pang and L. Zalcman, Normality and shared values, Ark. Mat. 38 (2000), 171–182. · Zbl 1079.30044 [10] X. C. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc. 32 (2000), 325–331. · Zbl 1030.30031 [11] X. C. Pang and L. Zalcman, On theorems of Hayman and Clunie, New Zealand J. Math. 28 (1999), 71–75. · Zbl 0974.30025 [12] W. Schwick, Sharing values and normality, Arch. Math. 59 (1992), 50–54. · Zbl 0758.30028 [13] Y. F. Wang and M. L. Fang, Picard values and normal families of meromorphic functions with zeros, Acta Math. Sinica (N.S.) 14 (1998), 17–26. · Zbl 0909.30025 [14] L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. 35 (1998), 215–230. · Zbl 1037.30021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.