# On certain subclasses of bounded univalent functions

Annales Polonici Mathematici (1991)

- Volume: 55, Issue: 1, page 109-115
- ISSN: 0066-2216

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topJ. 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 -

## References

top- [1] J. Fuka and Z. Jakubowski, On certain subclasses of bounded univalent functions, in: Proc. of the XI-th Instructional Conference on the Theory of Extremal Problems, Łódź, 1990, 20-27 (in Polish). Zbl0755.30023
- [2] A. I. Markushevich, Theory of Analytic Functions, Vol. 2, Nauka, Moscow 1968 (in Russian). Zbl0506.01013
- [3] P. T. Mocanu, Une propriété de convexité généralisée dans la théorie de la représentation conforme, Mathematica (Cluj) 11 (1969), 127-133. Zbl0195.36401
- [4] G. Pick, Über die konforme Abbildung eines Kreises auf ein schlichtes und zugleich beschränktes Gebiet, Sitzungsber. Akad. Wiss. Wien 126 (1917), 247-263. Zbl46.0553.01
- [5] K. Skalska, Certain subclasses of the class of typically real functions, Ann. Polon. Math. 38 (1980), 141-152. Zbl0465.30012

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