On conjecture of R.Brück concerning the entire function sharing one value CM with its derivative
For an entire function \(f\) the order \(\sigma(f)\) and the hyper order \(\sigma_2(f)\) are defined by \[ \sigma(f) = \limsup_{r\to\infty}{\frac{\log{T(r,f)}}{\log{r}}}\,, \quad \sigma_2(f) = \limsup_{r\to\infty}{\frac{\log{\log{T(r,f)}}}{\log{r}}}\,, \] where \(T(r,f)\) denotes the Nevanlinna characteristic of \(f\). Two non-constant entire functions \(f\) and \(g\) share a value \(a \in \mathbb{C}\) CM (counting multiplicities), if \(f-a\) and \(g-a\) have the same zeros with the same multiplicities. Then a conjecture of \textit{R.~Brück} [Result. Math. 30, 21--24 (1996; Zbl 0861.30032)] states that if a non-constant entire function \(f\) shares a value \(a \in \mathbb C\) CM with its derivative \(f'\), and if \(\sigma_2(f)<\infty\) is not a positive integer, then \(f-a=c(f'-a)\), where \(c\) is a nonzero constant. Brück already proved the conjecture for \(a=0\). It is also true under the additional assumption \(N(r,0,f')=S(r,f)\) without any growth restrictions. He also gave examples that it does not hold, if \(\sigma_2(f)=\infty\) or if \(\sigma_2(f)\) is a positive integer. For functions of finite order \(\sigma(f)\) the conjecture was proved by \textit{G.G.~Gundersen} and \textit{L.-Z.~Yang} [J. Math. Anal. Appl. 223, 88--95 (1998; Zbl 0911.30022)] (this paper is not mentioned in the references) and also by \textit{L.-Z.~Yang} [Kodai Math. J. 22, 458--464 (1999; Zbl 1004.30021)] using another method. In this paper the authors prove the conjecture for functions with \(\sigma_2(f)<\frac{1}{2}\). The main tools in the proof are the Hadamard factorization theorem and the central index.
scientific article; zbMATH DE number 2113506