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The Borel direction of the largest type of algebroid functions dealing with multiple values. (English) Zbl 1134.30023
Let $w=w(z)$ 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. 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.

30D35Distribution of values (one complex variable); Nevanlinna theory
30D30General theory of meromorphic functions
Full Text: DOI Euclid