Displaying similar documents to “Some applications of the method of extremal points in the theory of analytic functions of one complex variable”

Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions

Daniel Girela (1996)

Colloquium Mathematicae

Similarity:

A well known result of Beurling asserts that if f is a function which is analytic in the unit disc Δ = z ∈ ℂ : | z | < 1 and if either f is univalent or f has a finite Dirichlet integral then the set of points e i θ for which the radial variation V ( f , e i θ ) = ∫ 0 1 | f ' ( r e i θ ) | d r is infinite is a set of logarithmic capacity zero. In this paper we prove that this result is sharp in a very strong sense. Also, we prove that if f is as above then the set of points e i θ such that ( 1 - r ) | f ' ( r e i θ ) | ≠ o ( 1 ) as r → 1 is a set of logarithmic capacity zero. In particular, our...