On certain subclasses of bounded univalent functions

J. Fuka, and Z. J. Jakubowski. "On certain subclasses of bounded univalent functions." Annales Polonici Mathematici 55.1 (1991): 109-115. <http://eudml.org/doc/262312>.

@article{J1991,
abstract = {Let = z ∈ ℂ; |z| < 1, T = z ∈ ℂ; |z|=1. Denote by S the class of functions f of the form f(z) = z + a₂z² + ... holomorphic and univalent in , and by S(M), M > 1, the subclass of functions f of the family S such that |f(z)| < M in . We introduce (and investigate the basic properties of) the class S(M,m;α), 0 < m ≤ M < ∞, 0 ≤ α ≤ 1, of bounded functions f of the family S for which there exists an open $arc I_α = I_α(f) ⊂ T$ of length 2πα such that $\overline\{lim\}_\{z→ z₀, z ∈ \} |f(z)| ≤ M$ for every $z₀ ∈ I_α$ and $\overline\{lim\}_\{z→ z₀,z∈ \} |f(z)|≤ m$ for every $z₀ ∈ T \ Ī_α$.},
author = {J. Fuka, Z. J. Jakubowski},
journal = {Annales Polonici Mathematici},
keywords = {two constant theorem; harmonic measure; univalent; bounded functions; Pick's function; Koebe function; compactness; topology of uniform convergence},
language = {eng},
number = {1},
pages = {109-115},
title = {On certain subclasses of bounded univalent functions},
url = {http://eudml.org/doc/262312},
volume = {55},
year = {1991},
}

TY - JOUR
AU - J. Fuka
AU - Z. J. Jakubowski
TI - On certain subclasses of bounded univalent functions
JO - Annales Polonici Mathematici
PY - 1991
VL - 55
IS - 1
SP - 109
EP - 115
AB - Let = z ∈ ℂ; |z| < 1, T = z ∈ ℂ; |z|=1. Denote by S the class of functions f of the form f(z) = z + a₂z² + ... holomorphic and univalent in , and by S(M), M > 1, the subclass of functions f of the family S such that |f(z)| < M in . We introduce (and investigate the basic properties of) the class S(M,m;α), 0 < m ≤ M < ∞, 0 ≤ α ≤ 1, of bounded functions f of the family S for which there exists an open $arc I_α = I_α(f) ⊂ T$ of length 2πα such that $\overline{lim}_{z→ z₀, z ∈ } |f(z)| ≤ M$ for every $z₀ ∈ I_α$ and $\overline{lim}_{z→ z₀,z∈ } |f(z)|≤ m$ for every $z₀ ∈ T \ Ī_α$.
LA - eng
KW - two constant theorem; harmonic measure; univalent; bounded functions; Pick's function; Koebe function; compactness; topology of uniform convergence
UR - http://eudml.org/doc/262312
ER -