Let be a domain in . For a meromorphic function in and let
Two meromorphic functions and in are said to share the value if . A result of W. Schwick [Arch. Math. 59, 50-54 (1992; Zbl 0758.30028)] states that if is a family of meromorphic functions in such that and share three distinct values , , for every , then is normal in . The corresponding statement in which is replaced by 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 , all of whose zeros are of multiplicity at least . If there exist and such that for every , and for all , then is a normal family in .
The corresponding result for holomorphic functions with is due to X. Pang [Analysis, München 22, 175-182 (2002; Zbl 1030.30031)] and requires only and that for . In the special case , 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)].