The uniqueness of an entire function sharing a small entire function with its derivatives
The essential result in this paper is the following improvement to an earlier result due to \textit{G. G. Gundersen} and \textit{L.-Z. Yang} in [J. Math. Anal. Appl., 223, 88--95 (1998; Zbl 0911.30022)]: Let \(f\) be a non-constant entire function of finite lower order \(\mu(f)\), \(a\) be a non-vanishing entire function such that \(\rho(a)<\mu(f)\) and \(n\) be a natural number. If \(f\), \(f^{(n)}\), and \(f^{(n+1)}\) share \(a\) IM, while \(f^{(n)}\), \(f^{(n+1)}\) share \(a\) CM, then \(f=f'\). Actually, this result is being deduced here from preparatory order considerations of certain linear differential equations in the complex plane, typically of type \(f^{(k)}-a_{1}= (f-a_{2})e^{Q}\), where \(a_{1},a_{2}\) are entire functions of order \(<1\), and \(Q\) is a polynomial. In this situation, it will be proved for the hyper-order of \(f\) that \(\mu_{2}(f)=\rho_{2}(f)=\)deg \(Q\). This result for the hyper-order is more or less well known, while the assertion for the lower hyper-order seems to be new. The proofs in the paper essentially are based on a Wiman--Valiron type reasoning. (Reviewers remark: Some parts of the proofs could have been made shorter.)
scientific article; zbMATH DE number 5555205
scientific article