On Angular Limits of Normal Meromorphic Functions: A Geometric Aspect

We will prove the assertions which give necessary and sufficient conditions for a normal meromorphic function on the open unit disk to have an angular limit. The results obtained show that the conditions from the classical Lindel？f theorem, as well as the theorems of Lehto and Virtanen and Bagemihl and Seidel, concerning angular limit values of meromorphic functions, can be weakened. 1. Introduction and Preliminaries Let be the open unit disk in the complex plane , with the boundary , and let be the Riemann sphere. For any and a function , let , where ( ) is a M？bius map. Further, the pseudohyperbolic distance？？ on is given by The function defined on as is the hyperbolic metric on . The chordal metric on the Riemann sphere is defined as All the convergence in this paper will be considered with respect to some of the aforementioned metrics. Since the convergence with respect to the hyperbolic and pseudohyperbolic metrics on the disk is equivalent, in our proofs we will use one of these metrics that “simplifies” the related proof. For a fixed , the set defined as is called the pseudohyperbolic disk with the pseudohyperbolic center and the pseudohyperbolic radius . Notice that for , , . Similarly, for a fixed , the set defined as is called the hyperbolic disk with the center and the hyperbolic radius . For simplicity, here as always in the sequel, for an arbitrary nonempty set we write if a sequence of complex functions defined on the disk tends uniformly on to the function with respect to the pseudohyperbolic (or hyperbolic) metric (or ) and the chordal metric of the Riemann sphere . Given a set so that ( is the closure of ) and the function , denote by the cluster set of the function at the point with respect to the set . Namely, is the set of all points for which there exists a sequence in so that and as . It is known that . If is a Stolz angle of the disk with the vertex at the point , then the cluster set is the limit value of the function along the angle . If for each with we have for some , then is said to be a Fatou point of the function and is its angular limit value. Here, as always in the sequel, will denote a Jordan arc that ends at a point . If for some then is said to be an asymptotic value of the function at the point along the curve . The classical Lindel？f theorem on boundary values of holomorphic functions asserts that if a bounded analytic function on the disk has an asymptotic value , , at a point , then is its angular limit value and is a Fatou point of the function [1]. Seidel [2] and Seidel and Walsh [3] investigated the boundary