Analytic Functions with Large Sets of Fatou Points

• Published in: Proceedings of the American Mathematical Society 1984-01, Vol.90 (2), p.293-298
• Main Authors: Hwang, J. S., Lappan, Peter
• Format: Article
• Language: English
• Subjects: Analytic functions; Asymptotic value; Conformal mapping; Exact sciences and technology; Function theory, analysis; Integers; Mathematical functions; Mathematical methods in physics; Mathematical theorems; Physics; Rectifiable curves; Simply connected regions; Tangents
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-1984-0727253-5

For a function f analytic in the unit disc D, and for each$\lambda > 0$, let$L(\lambda) = \{z \in D: |f(z)| = \lambda \}$denote a level set for f. We introduce a class L1of functions characterized by geometric properties of a collection of sets$\{L(\lambda_n)\}$, where { λn} is an unbounded sequence. We show that L1is a proper subclass of the class L of G. R. MacLane. Let A∞denote the set of points eiθat which the function f has ∞ as an asymptotic value, and let F(f) denote the set of Fatou points of f. We prove that for a function f in the class L1, if Γ is an arc of the unit circle such that$\Gamma \cap A_\infty = \varnothing$, then almost every point of Γ belongs to F(f).