Koebe sets for certain classes of circularly symmetric functions

Paweł Zaprawa

doi:10.1016/j.crma.2015.12.016

Mathematical analysis/Complex analysis

Koebe sets for certain classes of circularly symmetric functions

Presented by: the Editorial Board

Paweł Zaprawa ¹

¹ Department of Mathematics, Lublin University of Technology, Nadbystrzycka 38D, 20-618 Lublin, Poland

Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 245-252.

Abstract
Résumé

A function f analytic in $Δ \equiv {ζ \in C : | ζ | < 1}$ , normalized by $f (0) = f^{'} (0) - 1 = 0$ , is said to be circularly symmetric if the intersection of the set $f (Δ)$ and a circle ${ζ \in C : | ζ | = ϱ}$ has one of three forms: the empty set, the whole circle, an arc of the circle which is symmetric with respect to the real axis and contains ϱ. By X we denote the class of all circularly symmetric functions, and by Y the subclass of X consisting of univalent functions.

The main concern of the paper is to determine two Koebe sets: for the class $Y \cap K (i)$ of circularly symmetric functions that are convex in the direction of the imaginary axis and for the class $Y \cap S^{⁎}$ of circularly symmetric and starlike functions, i.e. sets of the form $K_{Y \cap K (i)} = ⋂_{f \in Y \cap K (i)} f (Δ)$ and $K_{Y \cap S^{⁎}} = ⋂_{f \in Y \cap S^{⁎}} f (Δ)$ . In the last section of the paper, we consider a similar problem for the class $Y \cap S^{⁎} \cap K (i)$ .

Une fonction f analytique dans $Δ \equiv {ζ \in C : | ζ | < 1}$ , normalisée par $f (0) = f^{'} (0) - 1 = 0$ , est dite circulairement symétrique si l'intersection de l'ensemble $f (Δ)$ et d'un cercle ${ζ \in C : | ζ | = ρ}$ est, soit l'ensemble vide, soit le cercle complet, soit un arc de cercle symétrique par rapport à l'axe réel et contenant ρ. Nous notons X la classe des fonctions circulairement symétriques et Y la sous-classe de X des fonctions univalentes.

L'objet de cette Note est de déterminer les ensembles de Koebe pour la classe $Y \cap K (i)$ des fonctions circulairement symétriques qui sont convexes dans la direction de l'axe imaginaire et pour la classe $Y \cap S^{⁎}$ des fonctions circulairement symétriques qui sont étoilées, c'est-à-dire de déterminer les ensembles $K_{Y \cap K (i)} = ⋂_{f \in Y \cap K (i)} f (Δ)$ et $K_{Y \cap S^{⁎}} = ⋂_{f \in Y \cap S^{⁎}} f (Δ)$ . Dans la dernière section, nous considérons ce problème pour la sous-classe $Y \cap S^{⁎} \cap K (i)$ .

Received: 2015-03-09
Accepted: 2015-12-16
Published online: 2016-02-04

DOI: 10.1016/j.crma.2015.12.016

Author's affiliations:

Paweł Zaprawa ¹

¹ Department of Mathematics, Lublin University of Technology, Nadbystrzycka 38D, 20-618 Lublin, Poland

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     title = {Koebe sets for certain classes of circularly symmetric functions},
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     pages = {245--252},
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Paweł Zaprawa. Koebe sets for certain classes of circularly symmetric functions. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 245-252. doi : 10.1016/j.crma.2015.12.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.12.016/

References
Cited by

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