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On transcendental meromorphic functions with radially distributed values. (English) Zbl 1081.30032
Let $f(z)$ be a transcendental meromorphic function. For an unbounded subset $X$ of the complex plane $\bbfC$, the author denotes by $n(r,X,f =a)$ and $\overline n(r,X,f =a)$ the number of the roots repeated according to multiplicity and distinct roots of $f(z)- a= 0$, $a\in\overline{\bbfC}$, in $X\cap\{z: |z|< r\}$. The integrated counting functions $N(r,X,f = a)$ and $\overline N(r,X, f =a)$ are defined in the usual manner. The author considers $q$ pair $\{\alpha_j,\beta_j\}$ of real numbers such that $-\pi\le \alpha_1< \beta_1\le\alpha_2< \beta_2\le\cdots\le \alpha_q< \beta_q\le\pi$ and defines $\omega= \max_j\{\pi/(\beta_j- \alpha_j)\}$. The author obtains several theorems in this paper. One of the main result is the following. Let $f(z)$ be a transcendental meromorphic function of finite lower order $\mu$. Suppose that $f(z)$ satisfies $\delta:=\delta(a,f^{(p)})> 0$ for some $a\in\overline{\bbfC}$ and an integer $p$. If for $q$ pair $\{\alpha_j, \beta_j\}$ of real numbers given above and for an integer $k> 0$, it holds $$n(r,Y,f= 0)+\overline n(r,Y,f^{(k)}= 1)= o(T(dr, f)),\quad d\ge 1,$$ for $Y= \bigcup^q_{j=1} \{z: \alpha\le\arg z\le\beta_j\}$ and $\sum^q_{j=1} (\alpha_{j+1}- \beta_j)< (4/\beta)\arcsin \sqrt{\delta/2}$, $\alpha_{q+1}= 2\pi+ \alpha_1$, where $\beta= \max\{\omega,\mu\}$, then $\lambda(f)\le\omega$. The methods for the proofs are from the Nevanlinna theory in the angular domains, and Baernstein’s theorem on the spread relation. The author comments on a singular direction in terms of the Nevanlinna characteristic function. A radial $\arg z= 0$ is called $T$ direction of $f(z)$ provided that given any $b\in\overline{\bbfC}$, possibly with the exception of at most two values for arbitrary small $\varepsilon> 0$ it holds $$\limsup_{r\to\infty} {N(r,Z,f= b)\over T(r,f)}> 0,$$ where $Z= \{z:\theta- \varepsilon< \arg z<\theta+\varepsilon\}$.

30D35Distribution of values (one complex variable); Nevanlinna theory
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