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Normality and shared values. (English) Zbl 1079.30044
For f meromorphic on the unit disc Δ and a define E ¯ f (a)=f -1 ({a})Δ={zΔ:f(z)=a}. Two functions f and g on Δ are said to share the value a if E ¯ f (a)=E ¯ g (a). A meromorphic function f on  is called a normal function if there exists a positive number M such that f # (z)M, where f # (z)=|f ' (z)|/(1+|f(z)| 2 ) denotes the spherical derivative. The authors prove the following theorems: 1) Let be a family of meromorphic functions on the unit disc Δ, and let a and b be distinct complex numbers and c a nonzero complex number. If for every f, E ¯ f (0)=E ¯ f ' (a), E ¯ f (c)=E ¯ f ' (b), then is normal on Δ. Earlier a similar result has been proved by W. Schwick [Arch. Math. 59, No. 1, 50–54 (1992; Zbl 0758.30028)]. 2) Let f be a meromorphic function on  and a and b be distinct complex numbers. If f and f ' share a and b, then f is a normal function. This should be compared with the result of E. Mues and N. Steinmetz in [Manuscr. Math. 29, 195–206 (1979; Zbl 0416.30028)].

MSC:
30D45Bloch functions, normal functions, normal families
References:
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[8]Mues, E. andSteinmetz, N., Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen,Manuscripta Math. 29 (1979), 195–206. · Zbl 0416.30028 · doi:10.1007/BF01303627
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[10]Pang, X., Shared values and normal families, Preprint, 1998.
[11]Pang, X. andZalcman, L., Normal families and shared values, to appear inBull. London Math. Soc.
[12]Schwick, W., Sharing values and normality,Arch. Math. (Basel) 59 (1992), 50–54.
[13]Yang, L.,Value Distribution Theory, Springer-Verlag, Berlin-Heidelberg, 1993.