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Normal families and shared values. (English) Zbl 1030.30031

Let D be a domain in . For a meromorphic function f in D and a let

E ¯ f (a)={zD:f(z)=a}·

Two meromorphic functions f and g in D are said to share the value a if E ¯ f (a)=E ¯ g (a). A result of W. Schwick [Arch. Math. 59, 50-54 (1992; Zbl 0758.30028)] states that if is a family of meromorphic functions in D such that f and f ' share three distinct values a 1 , a 2 , a 3 for every f, then is normal in D. The corresponding statement in which f ' is replaced by f (k) (k2) is no longer true. A counterexample was given by G. Frank and W. Schwick [N. Z. J. Math. 23, 121-123 (1994; Zbl 0830.30019)]. In this paper the authors prove the following result.

Theorem. Let be a family of meromorphic functions in D, all of whose zeros are of multiplicity at least k. If there exist b{0} and h>0 such that for every f, E ¯ f (0)=E ¯ f (k) (b) and 0<|f (k+1) (z)|h for all zE ¯ f (0), then is a normal family in D.

The corresponding result for holomorphic functions with k=1 is due to X. Pang [Analysis, München 22, 175-182 (2002; Zbl 1030.30031)] and requires only E ¯ f (0)E ¯ f ' (b) and that |f '' (z)|h for zE ¯ f ' (b). In the special case E ¯ f (0)=, the above theorem gives a result of Y. Ku [Sci. Sinica 1979, Special Issue I on Math., 267-274 (1979)].

In contrast to the proofs of the above results of X. Pang and W. Schwick, the authors make no use of Nevanlinna theory. The main tool of the proof is a generalization of a version of the non-normality criterion of Z. Zalman [Am. Math. Mon. 82, 813-817 (1975; Zbl 0315.30036)] which is due to X. Pang [Sci. China, Ser. A 32, 782-791 (1989; Zbl 0687.30023)], [Sci. China, Ser. A 33, 521-527 (1990; Zbl 0706.30024)].


MSC:
30D45Bloch functions, normal functions, normal families
30D35Distribution of values (one complex variable); Nevanlinna theory