On uniqueness of meromorphic functions with shared values in one angular domain. (English) Zbl 1041.30009

Zheng, Jian-Hua

Complex Variables, Theory Appl. 48, No. 9, 777-785 (2003).

Let \(f(z)\) and \(g(z)\) be transcendental meromorphic functions in the complex plane. A value \(a\in \widehat{\mathbb C}= \mathbb C\cup\{\infty\}\) is called an IM (ignoring multiplicities) shared value in \(X\subset \mathbb C\) of \(f(z)\) and \(g(z)\) if in \(X\), \(f(z)= a\) if and only if \(g(z)= a\). A value \(a\in \widehat{\mathbb C}\) is called an CM (counting multiplicities) shared value in \(X\subset \mathbb C\) of \(f(z)\) and \(g(z)\) if in \(X\), \(f(z)\) and \(g(z)\) assume \(a\) at the same points in \(X\) with the same multiplicities. In the case \(X= \mathbb C\), there are several sharing conditions for uniqueness, see e.g., G. G. Gundersen [J. Lond. Math. Soc., II. Ser. 20, 456–466 (1979; Zbl 0413.30025)]. For example, (C1) If \(f(z)\) and \(g(z)\) share five values IM, then \(f(z)\equiv g(z)\), which is due to R. Nevanlinna. (C2) If \(f(z)\) and \(g(z)\) share four values IM, and if for another value \(\delta(a,f)> 0\), then \(f(z)\equiv g(z)\). (C3) If \(f(z)\) and \(g(z)\) share two values IM and if \(f(z)\) and \(g(z)\) share two values CM, then \(f(z)\) and \(g(z)\) share four values of CM. The author considers the uniqueness problem with the sharing conditions in some angular domains.
In this paper three theorems are obtained generalizing the (C1), (C2) and (C3). We state one of the results. Given an angular domain \(X= \{z: \alpha<\arg z< \beta\}\) with \(0\leq \alpha< 1\beta\leq 2\pi\) and for some positive number \(\varepsilon\) and for some \(a\in \widehat{\mathbb C}\), \(\limsup_{r\to\infty} \log n(r, X_\varepsilon, f= a)/\log r> \omega\), where \(n(r, X_\varepsilon, f=a)\) is the number of the roots of \(f(z)= a\) in \(\{| z|< r\}\cap X_\varepsilon\), \(X_\varepsilon= \{z: \alpha+ \varepsilon< \arg z<\beta- \varepsilon\}\) and \(\omega= \pi/(\beta- \alpha)\). If \(f(z)\) and \(g(z)\) share five distinct values IM, then \(f(z)\equiv g(z)\).

Reviewer: Katsuya Ishizaki (Saitama)

MSC:

30D35

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

Keywords:

Nevanlinna theory; meromorphic function; shared value

Citations:

Zbl 0413.30025