On a problem of Gross concerning unicity of meromorphic functions. (Chinese) Zbl 0889.30028

Yi Hongxun [Sci. China, Ser. A 37, No. 7, 802-813 (1994; Zbl 0821.30024)] has proved three theorems in order to solve the problem posed by F. Gross [Complex Anal., Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 51-67 (1976; Zbl 0357.30007)].
In this paper, the authors prove some analogous theorems with Yi’s theorems and obtain that if \(f\), \(g\) satisfy one of the following conditions: 1) \(f\), \(g\) are two non-constant meromorphic functions and \(\overline E_f(S)=\overline E_g(S)\), \(n>14\); 2) \(f\), \(g\) are two non-constant entire functions and \(\overline E_f(S)=\overline E_g(S)\), \(n>7\); 3) \(f\), \(g\) are two non-constant meromorphic functions and \(\overline E_f(\widetilde S)=\overline E_g(\widetilde S)\), \(\overline E_f(S)=\overline E_g(S)\), \(n>13\), then \(f-a= v(g- a)\), \(v^n=1\) or \((f- a)(g- a)= b^2u\), \(u^n= 1\), where \(n\in\mathbb{N}\), \(a(\neq 0)\), \(b(\neq 0)\in \mathbb{C}\), \(w= \cos{2\pi\over n}+ i\sin{2\pi\over n}\), \(S= \{a+ b,a+ bw,\dots, a+bw^{n+ 1}\}\), \(\widetilde S= \{a\}\) or \(\{\infty\}\), \(\overline E_f(S)= \bigcup_{\sigma\in S} \{z\in\widehat C\mid f(z)= c\}\).


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable