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Uniqueness and value-sharing of entire functions. (English) Zbl 1152.30034
The paper under review deals with unicity theorems for entire functions. Let $$f$$ and $$g$$ be non-constant entire functions on $$\mathbb{C}$$. The main theorem in this paper can be stated as follows: Let $$\mu$$ and $$\lambda$$ be constants such that $$|\mu|+ |\lambda|\neq 0$$. Let $$n$$, $$m$$ and $$k$$ be positive integers with $$n> 2k+ m^*+ 4$$, where $$m^*= 0$$ if $$\mu= 0$$ and $$m^*= m$$ if $$\mu\neq 0$$. Suppose that $$[f^n(z)(\mu f^m(z)+ \lambda)]^{(k)}$$ and $$[g^n(z)(\mu g^m(z)+ \lambda)]^{(k)}$$ share 1 CM. If $$\mu\lambda\neq 0$$, then $$f\equiv g$$. If $$\mu\lambda= 0$$, then either $$f\equiv tg$$ or $$f(z)= c_1 e^{cz}$$, $$g(z)= c_2 e^{cz}$$, where $$t$$, $$c$$, $$c_1$$ and $$c_2$$ are constants depending on $$k$$, $$m$$, $$n$$, $$\mu$$ and $$\lambda$$. The authors also prove the following: If $$n> 2k+ m+ 4$$ and $$[f^n(z)(\mu f^m(z)- 1)]^{(k)}$$ and $$[g^n(z)(\mu g^m(z)- 1)]^{(k)}$$ share 1 CM, then either $$f\equiv g$$ or $$f$$ and $$g$$ satisify the algebraic relation $$R(f,g)\equiv 0$$, where $$R(w_1,w_2)= w^n_1(w_1- 1)^m- w^n_2(w_2- 1)^m$$.

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