Algebroid functions with given Borel points

Abstract

Abstract In this paper, we prove the following result. Suppose that E is an arbitrary non-empty closed set of real numbers ( mod 2 π ) ${(\operatorname{mod} 2\pi )}$ . Then for any positive λ ( 0 < λ < ∞ ${0<\lambda <\infty }$ ), there exists an algebroid function (not meromorphic) w(z) of order λ in the unit disc, such that every point e i θ ${e^{i\theta }}$ , θ ∈ E ${\theta \in E}$ , on the unit circle is a Borel point of order λ of w(z) and w(z) has no other Borel point of order λ. This result generalizes the related results of a meromorphic function due to Lo Yang, Guanghou Zhang, Daochun Sun and Zongsheng Gao.