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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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