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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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