# Covering Problems for Functions $$n$$ -Fold Symmetric and Convex in the Direction of the Real Axis II

Bulletin of the Malaysian Mathematical Sciences Society, Oct 2015

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 $$z\in \Delta$$, where $$\varepsilon =e^{2\pi i/n}$$. The functions of this subclass are called $$n$$-fold symmetric. For $${\mathcal {F}}^{(n)}$$, where $$n$$ is odd positive integer, the following sets, $$\bigcap _{f\in {\mathcal {F}}^{(n)}} f(\Delta )$$—the Koebe set and $$\bigcup _{f\in {\mathcal {F}}^{(n)}} f(\Delta )$$—the covering set, are discussed. As corollaries, we derive the Koebe and the covering constants for $${\mathcal {F}}^{(n)}$$.

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Leopold Koczan, Paweł Zaprawa. Covering Problems for Functions $$n$$ -Fold Symmetric and Convex in the Direction of the Real Axis II, Bulletin of the Malaysian Mathematical Sciences Society, 2015, 1637-1655, DOI: 10.1007/s40840-014-0107-8