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Meromorphic functions sharing one value. (English) Zbl 1093.30024
We discuss the uniqueness problem of meromorphic functions sharing one value and obtain two theorems which improve a result of Xu and Qu and supplement some other results earlier given by Yang, Hua, and Lahiri. Let \(f\) and \(g\) be meromorphic functions in the complex plane, and let \(n\) be an integer. The author considers uniqueness problems of meromorphic functions with some conditions in \(f^nf'\) and \(g^ng'\). In this paper the author obtains two main results. Here the reviewer mentions one of them. Suppose that \(f\) and \(g\) satisfy \[ n>22-5\bigl(\Theta(\infty,f)+\Theta(\infty,g)\bigr)-\min\bigl\{\Theta (\infty,f),\Theta(\infty,g)\bigr\}. \] If for \(a\in\mathbb{C}\setminus\{0\}\), \(f^nf'\) and \(g^ng'\) share \(a\) IM, then either \(f=dg\) for some \(n+1\)-th root of the unity of \(d\) or \(g(z)=c_1e^{cz}\) and \(f(z)=c_2e^{-cz}\), where \(c\), \(c_1\), and \(c_2\) are constants satisfying \((c_1c_2)^{n+1}c^2=-a^2\). This result is an improvement for the theorem in [Y. Xu and H. Qu, Entire functions, sharing one value IM. Indian J. Pure Appl. Math. 31, No. 7, 849–855 (2000; Zbl 0964.30015)]. The main tools of the proofs are the value distribution theory.

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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