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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)

MSC:
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
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[2] E. MUES and N. STEINMETZ, Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen, Manuscpta Math, 29 (1979), 195-206 · Zbl 0416.30028 · doi:10.1007/BF01303627 · eudml:154660
[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
[4] HoNG-XuN Yi, Uniqueness of meromorphic functions and a question of C. C. Yang, Complex Vaables Theory Appl, 14 (1990), 169-176 · Zbl 0701.30025
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