- Tytuł:
- Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions
- Autorzy:
- Girela, Daniel
- Powiązania:
- https://bibliotekanauki.pl/articles/965130.pdf
- Data publikacji:
- 1996
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
radial variation
Dirichlet integral
capacity
univalent functions - Opis:
- 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 $e^{iθ}$ for which the radial variation $V(f,e^{iθ})=∫_{0}^{1}|f'(re^{iθ})|dr$ 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 $e^{iθ}$ such that $(1 - r)|f'(re^{iθ})| ≠ o(1)$ as r → 1 is a set of logarithmic capacity zero. In particular, our results give an answer to a question raised by T. H. MacGregor in 1983.
- Źródło:
-
Colloquium Mathematicum; 1996, 69, 1; 19-17
0010-1354 - Pojawia się w:
- Colloquium Mathematicum
- Dostawca treści:
- Biblioteka Nauki
Artykuł