Univalent harmonic mappings with two preassigned values

• Autorzy: Gregorczyk M.
• Języki publikacji: EN

Abstrakty

EN

In this paper we consider a class H*(a) of normalized harmonic functions which map the unit disk onto starlike domains. We give necessary and sufficient condition for is an element H*(a) distortion bounds and extreme points. We partially solve the problem of the Koebe set for H*(a). We also consider the Schild's Conjecture for the class TH* of harmonic mappings.

Słowa kluczowe

EN

harmonic mappings; Montel type normalization; Schild's conjecture; Koebe set

PL

odwzorowanie harmoniczne; normalizacja typu Montel; hipoteza Schiltza; zbiór Koebe

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2009
• Tom: Vol. 42, nr 4
• Strony: 711--722
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Gregorczyk M.

• Institute of Mathematics Maria Curie-Skłodowska University Pl. M. Curie-Skłodowska 1, 20-031 Lublin,

Bibliografia

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• [2] J. Clunie, T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I. Math. 9 (1984), 3-25.
• [3] P. Duren, Harmonic Mappings, Cambridge University Press, 2004.
• [4] E. Gray, A. Schild, A new proof of a conjecture of Schild, Proc. Amer. Math. Soc. 16 (1963), 76-77.
• [5] Z. Lewandowski, Nouvelles remarques sur les theoremes de Schild relatifs a une classe de fonctions univalentes (Demonstration d’une hypothese de Schild), Ann. Univ. Mariae Curie-Skłodowska Sect. A 8 (1956), 81-94.
• [6] A. Schild, On a class od functions schlicht in the unit circle, Proc. Amer. Math. Soc. 5 (1953), 115-120.
• [7] H. Silverman, Survey with open problems on univalent functions whose coefficients are negative, Rocky Mountain J. Math. 3 (1991), 1099-1125.