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A counterexample to the generalized Bloch principle. (English) Zbl 0830.30019
The authors present an interesting and (at least to the reviewer) surprising counterexample to the so-called Bloch principle for normal families. This principle – that properties which render a function which is meromorphic in the plane constant also yield criteria for normal families in the disk – is known not to hold in full generality, but it has been of great use in the study of normal families and Picard properties (some counterexamples to the principle are given in the references here, but an important success of it occurs in the recent work of W. Bergweiler and A. E. Eremenko, Mat. Iberoam. 11, No. 2, 355-373 (1995; reviewed above). The authors take another example of this principle as starting point. Functions $$f$$ and $$g$$ are said to share the value $$a$$ if $$f(z) = a$$ has the same solutions as $$g(z) = a$$. It is known that the result: if $$f$$ is meromorphic in the plane and $$f$$ and $$f'$$ share three complex numbers, then $$f \equiv f'$$, has a direct normal- family analogue. Recently, the authors obtained one part of this pairing for higher derivatives: if $$f$$ and $$f^{(k)}$$ share three values for some $$k \geq 2$$, then $$f \equiv f^{(k)}$$. The surprise is that this theorem, as well as some other alternative formulations, fail to have normal family analogues. The examples are based on the functions $$f_n (z) = n\{e^z - e^{\alpha z}\}$$, where $$\alpha$$ is an appropriate root of unity and the $$\{a_i\}$$ may be taken at him.

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