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Search: authors:"Leopold Koczan"

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Sets of Univalence in Some Classes of Analytic Functions

This paper attempts to determine the “largest” sets of local univalence for a given class and the “largest” open sets in which all functions belonging to a given class are univalent. We establish general properties of sets of univalence for analytic functions with typical normalization. Moreover, we determine some examples of the sets of univalence for some particular subclasses...

Circularly Symmetric Locally Univalent Functions

Let \(D\subset \mathbb {C}\) and \(0\in D\). A set D is circularly symmetric if, for each \(\varrho \in \mathbb {R}^+\), a set \(D\cap \{\zeta \in \mathbb {C}:|\zeta |=\varrho \}\) is one of three forms: an empty set, a whole circle, a curve symmetric with respect to the real axis containing \(\varrho \). A function f analytic in the unit disk \(\Delta \equiv \{\zeta \in \mathbb...

Covering Problems for Functions \(n\) -Fold Symmetric and Convex in the Direction of the Real Axis II

Let \({\mathcal {F}}\) denote the class of all functions univalent in the unit disk \(\Delta \equiv \{\zeta \in {\mathbb {C}}\,:\,\left| \zeta \right| <1\}\) and convex in the direction of the real axis. The paper deals with the subclass \({\mathcal {F}}^{(n)}\) of these functions \(f\) which satisfy the property \(f(\varepsilon z)=\varepsilon f(z)\) for all \(z\in \Delta...