On Angular Limits of Normal Meromorphic Functions: A Geometric Aspect

Zarko Pavicevic

doi:10.1155/2014/216398

Abstract

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 Lindelöf theorem, as well as the theorems of Lehto and Virtanen and Bagemihl and Seidel, concerning angular limit values of meromorphic functions, can be weakened.

Highlights

Introduction and PreliminariesLet D = {z ∈ C : |z| < 1} be the open unit disk in the complex plane C, with the boundary Γ = {z ∈ C : |z| = 1}, and let Ω = C = C ∪ {∞} be the Riemann sphere
The pseudohyperbolic distance dph on D is given by dph (z1, z2)
We prove Theorem 6 of Section 3 and Theorems 7–10 of Section 4, which give necessary and sufficient conditions for a normal meromorphic function f in D at a point eiθ to have an angular limit in terms of a sequence (f(zn)) with ⊂ D and limn → ∞zn = eiθ under the condition sup{dh(zn, zn+1) : n ∈ N} ≤ M < +∞

Summary

Introduction and Preliminaries

We prove Theorem 6 of Section 3 and Theorems 7–10 of Section 4, which give necessary and sufficient conditions for a normal meromorphic function f in D at a point eiθ to have an angular limit in terms of a sequence (f(zn)) with (zn) ⊂ D and limn → ∞zn = eiθ under the condition sup{dh(zn, zn+1) : n ∈ N} ≤ M < +∞. In Theorem 7 these sets are sufficiently large disks whose pseudohyperbolic (hyperbolic) centers are the terms of the sequence (zn) (see Figure 1); in Theorem 8, these sets are sufficiently large parts of Jordan’s arcs, and in Theorem 10 they are many sequences of points These theorems give new criteria for the existence of angular limit values of meromorphic functions on D at points eiθ of the unit circle Γ. These lemmas present auxiliary results for proofs of other assertions in this paper

Auxiliary Lemmas

The Main Result

Applications