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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 ν-valued algebroid function defined by an irreducible equation A ν (z)w ν +A ν-1 (z)w ν-1 ++A 0 (z)=0, where A k are entire functions without any common zeros. Assume that U(r)=r ρ(r) is the type function of w(z). A ray B θ ={z:argz=θ} (0θ<2π) is called a Borel direction of the largest type dealing with multiple values of w(z) if, for any ϵ>0 and any integer l2ν+1, lim sup r n ¯ l (r,Δ(B θ ,ϵ),a)/U(r)>0 holds for any complex value a except at most 2ν possible exceptions.

In this paper, the authors prove that if a ν-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.

MSC:
30D35Distribution of values (one complex variable); Nevanlinna theory
30D30General theory of meromorphic functions