The uniqueness of meromorphic functions with their derivatives
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 \textit{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 \(f^{(k)}\).
scientific article; zbMATH DE number 1269766