On certain subclasses of bounded univalent functions

• Autorzy: J. Fuka , Z. J. Jakubowski
• Języki publikacji: EN

Abstrakty

EN

Let 𝔻 = {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 𝔻, and by S(M), M > 1, the subclass of functions f of the family S such that |f(z)| < M in 𝔻. 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 $\overline{lim}_{z→ z₀, z ∈ 𝔻 } |f(z)| ≤ M$ for every $z₀ ∈ I_α$ and $\overline{lim}_{z→ z₀,z∈ 𝔻 } |f(z)|≤ m$ for every $z₀ ∈ T \ Ī_α$.

Kategorie tematyczne

• 30C55: General theory of univalent and multivalent functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1991
• Tom: 55
• Numer: 1
• Strony: 109-115

Daty

wydano

1991

otrzymano

1990-08-16

Twórcy

autor

J. Fuka

• Matematický Ústav ČSAV, Žitná 25, 11567 Praha 1, Czechoslovakia

autor

Z. J. Jakubowski

• Institute of Mathematics, Łódź University, Banacha 22, 90-238 Łódź, Poland

Bibliografia

• [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).
• [2] A. I. Markushevich, Theory of Analytic Functions, Vol. 2, Nauka, Moscow 1968 (in Russian).
• [3] P. T. Mocanu, Une propriété de convexité généralisée dans la théorie de la représentation conforme, Mathematica (Cluj) 11 (1969), 127-133.
• [4] G. Pick, Über die konforme Abbildung eines Kreises auf ein schlichtes und zugleich beschränktes Gebiet, Sitzungsber. Akad. Wiss. Wien 126 (1917), 247-263.
• [5] K. Skalska, Certain subclasses of the class of typically real functions, Ann. Polon. Math. 38 (1980), 141-152.