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The Borel direction of the largest type of algebroid functions dealing with multiple values. (English) Zbl 1134.30023

Let $w=w\left(z\right)$ be a $\nu$-valued algebroid function defined by an irreducible equation ${A}_{\nu }\left(z\right){w}^{\nu }+{A}_{\nu -1}\left(z\right){w}^{\nu -1}+\cdots +{A}_{0}\left(z\right)=0$, where ${A}_{k}$ are entire functions without any common zeros. Assume that $U\left(r\right)={r}^{\rho \left(r\right)}$ is the type function of $w\left(z\right)$. A ray ${B}_{\theta }=\left\{z:argz=\theta \right\}$ ($0\le \theta <2\pi$) is called a Borel direction of the largest type dealing with multiple values of $w\left(z\right)$ if, for any $ϵ>0$ and any integer $l\ge 2\nu +1$, ${lim sup}_{r\to \infty }{\overline{n}}^{l}\left(r,{\Delta }\left({B}_{\theta },ϵ\right),a\right)/U\left(r\right)>0$ holds for any complex value $a$ except at most $2\nu$ possible exceptions.

In this paper, the authors prove that if a $\nu$-valued algebroid function $w\left(z\right)$ 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.

##### MSC:
 30D35 Distribution of values (one complex variable); Nevanlinna theory 30D30 General theory of meromorphic functions
##### Keywords:
Borel direction; algebroid function; filling disk