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On uniqueness of meromorphic functions in an angular domain. (English) Zbl 1145.30012
Let $f$ be a transcendental meromorphic function, and let $\rho(r)$ be a continuous and nondecreasing function tends to $\infty$ as $r\to\infty$. We denote by $M(\rho(r))$ the set of all meromorphic functions satisfying $\limsup_{r\to\infty}\log T(r,f)/\log U(r)= 1$, where $U(r)= e^{\rho(r)}$. The author considers the uniqueness problem on the class $M(\rho(r))$ defined by a proximate order with some angular conditions. Concerning the proximate order, see e.g., {\it C. Chuang} [Sci. Sin. 10, 171--181 (1961; Zbl 0102.04704)]. Let $X$ be a domain in $\bbfC$, and $S$ be a set of complex numbers. Denote $E_X(S, f)=\bigcup_{a\in S}\{z\in\overline X|f(z)= a\}$. The author defines $S_1= \{0\}$, $S_2= \{\infty\}$ and $S_3= \{w| w^n(w+ a)- b= 0\}$, where $n\in\bbfN$ and the algebraic equation $w^n(w+a)- b= 0$ has nomultiple roots. It is mentioned that the following is proved. Let $f,g\in M(\rho(r))$ and suppose that $\delta(\infty, f)> 0$. Let $X= \{z|\arg z-\theta|< \varepsilon\}$, where $\varepsilon> 0$ and $0\le\theta< 2\pi$. Suppose that $n\ge 3$ in the definition of $S_3$, and for some $a\in\bbfC$, $$\limsup_{r\to\infty}\, \log n(r,\theta, \varepsilon/3, a)/\log U(r)= 1.$$ If $E_{\bbfC}(S_1, f)= E_{{\bbfC}}(S_1, g)$ and $E_X(S_j, f)= E_X(S_j, g)$, $j= 2,3$, then $f\equiv g$.
30D35Distribution of values (one complex variable); Nevanlinna theory
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