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Uniqueness and value-sharing of entire functions. (English) Zbl 1035.30017
Summary: We study the uniqueness of entire functions and prove the following theorem. Let $$f(z)$$ and $$g(z)$$ be two nonconstant entire functions, $$n, k$$ two positive integers with $$n > 2k + 4$$. If $$[f^n(z)]^{(k)}$$ and $$[g^n(z)]^{(k)}$$ share 1 with counting the multiplicity, then either $$f(z) = c_1e^{cz}$$, $$g(z) = c_2e^{-cz}$$, where $$c_1$$, $$c_2$$, and $$c$$ are three constants satisfying $$(-1)^k(c_1c_2)^n(nc)^{2k} = 1$$, or $$f(z)\equiv tg(z)$$ for a constant $$t$$ such that $$t^n = 1$$.

MSC:
 30D20 Entire functions of one complex variable (general theory) 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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