# 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