The value distribution of \(f^ n f'\). (English) Zbl 0839.30026

The Hayman’s conjecture “If \(f\) is a transcendental meromorphic function, then \(f^nf\) assumes every finite non-zero complex value infinitely often for any positive integer \(n\)” has been completely proved.
The authors prove (theorem 1) the Hayman’s conjecture for \(n= 1\). Together with earlier investigations of other authors it proves the Hayman’s conjecture.
Other results are new criteria for the normality of the family of meromorphic functions (theorem 2, 3) and strengthen variants (it is proved an existence of Julia’s ray) of the theorem 1 and the Hayman’s conjecture (theorem 4, 5) under the assumptions \[ \varlimsup_{r\to \infty} {T(r, f)\over (\ln r)^2}= \infty,\quad \varlimsup_{r\to \infty} {T(r, f)\over (\ln r)^2}= \infty. \] The authors state that W. Bergweiler and A. Eremenko, and L. Zalcman also proved the Hayman’s conjecture simultaneously and independently.


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory