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Zbl 1134.30023
Xuan, Zu-Xing; Gao, Zong-Sheng
The Borel direction of the largest type of algebroid functions dealing with multiple values. (English)
[J] Kodai Math. J. 30, No. 1, 97-110 (2007). ISSN 0386-5991

Let $w=w(z)$ be a $\nu$-valued algebroid function defined by an irreducible equation $A_{\nu}(z)w^{\nu}+A_{\nu-1}(z)w^{\nu-1}+\cdots +A_{0}(z)=0$, where $A_{k}$ are entire functions without any common zeros. Assume that $U(r)=r^{\rho(r)}$ is the type function of $w(z)$. A ray $B_{\theta}=\{z: \arg z =\theta\}$ ($0\le \theta < 2 \pi$) is called a Borel direction of the largest type dealing with multiple values of $w(z)$ if, for any $\epsilon>0$ and any integer $l\ge 2\nu +1$, $\limsup_{r \to \infty}\bar{n}^l(r, \Delta(B_{\theta}, \epsilon), a )/U(r) >0 $ holds for any complex value $a$ except at most $2\nu$ possible exceptions. \par In this paper, the authors prove that if a $\nu$-valued algebroid function $w(z) $ is of finite positive order then there exists a Borel direction of the largest type dealing with multiple values, and moreover, there is a sequence of filling disks in this direction.
[Zhuan Ye (DeKalb)]

MSC 2000:: *30D35 Distribution of values (one complex variable)
30D30 General theory of meromorphic functions

Keywords: Borel direction; algebroid function; filling disk

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