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On Mues conjecture and Picard values. (English) Zbl 0777.30017
Let $$f$$ be a transcendental meromorphic function. Then
(i) $$\sum_{a\in\mathbb{C}}\delta(a,f^{(k)})\leq 1$$ holds for all $$k\geq 0$$ with at most four exceptions.
(ii) For $$n\geq 3$$, $$k\geq 0$$ $$(f^ n)^{(k)}$$ assumes every complex value other than 0 infinitely often. Improvements of W. K. Hayman’s bound on $$T(r,f)$$ in terms of $$N(r,1/f)$$ and $$N(r,1/(f^{(k)}-1))$$ [Meromorphic functions (1964; Zbl 0115.062)] are also given. These results are important improvement of previous work by many authors.
Reviewer: W.H.Fuchs (Ithaca)

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
##### Keywords:
Mues conjecture; Picard values