Uniqueness and value-sharing of entire functions. (English) Zbl 1152.30034

Zhang, Xiao-Yu; Lin, Wei-Chuan

J. Math. Anal. Appl. 343, No. 2, 938-950 (2008).

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 \(R(w_1,w_2)= w^n_1(w_1- 1)^m- w^n_2(w_2- 1)^m\).

Reviewer: Yoshihiro Aihara (Shizuoka)

MSC:

30D35

Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

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