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The uniqueness of meromorphic functions with their derivatives. (English) Zbl 0932.30027
The paper deals with the question up to which class of functions a meromorphic function \(f\) in \(\mathbb{C}\) is determined if it shares with its derivative \(f'\) the value 1 \(CM\) (= counting multiplicity). This means that the preimages of 1 under \(f\) and \(f'\) are equal and the 1-points of \(f\) and \(f'\) have the same multiplicity. The main result obtained is that if in addition \[ \overline N(r,f)+N \left(r,{1\over f'}\right) <\bigl(\lambda+ o(1)\bigr) T(r,f') \] \((N,T\) from Nevanlinna-theory), for some \(\lambda\in(0,{1\over 2})\) then \[ f(z)=Ae^{cz}+ 1-{1\over c}, \quad A,c\in \mathbb{C}\setminus \{0\}. \] This generalizes a result of R. Brück [Result Math. 30, No. 1-2, 21-24 (1996; Zbl 0861.30032)]. There are also obtained similar results if either \(f\) is entire or \(f'\) is replaced by some higher derivative \(f^{(k)}\).
Reviewer: G.Jank (Aachen)

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable, general theory
30D20 Entire functions of one complex variable, general theory
Full Text: DOI
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[3] RAINER BRUCK, On entire functions which share one value CM with their first derivative, Results in Math, 30 (1996), 21-24 · Zbl 0861.30032 · doi:10.1007/BF03322176
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