Turkish Journal of Mathematics
DOI
10.3906/mat-1511-31
Abstract
In this paper, we consider a class $K_\alpha$ of all functions $f$ univalent in the unit disk $\Delta$ that are normalized by $f(0)=f'(0)-1=0$ while the sets $f(\Delta)$ are convex in two symmetric directions: $e^{i\alpha\pi/2}$ and $e^{-i\alpha\pi/2}$, $\alpha\in[0,1]$. It means that the intersection of $f(\Delta)$ with each straight line having the direction $e^{i\alpha\pi/2}$ or $e^{-i\alpha\pi/2}$ is either a compact set or an empty set. We find the Koebe set for $K_\alpha$. Moreover, we perform the same operation for functions in $K_{\beta, \gamma}$, i.e. for functions that are convex in two fixed directions: $e^{i\beta\pi/2}$ and $e^{i\gamma\pi/2}$, $-1\leq \beta\leq\gamma \leq 1$.
Keywords
Univalent functions, convexity in a direction
First Page
282
Last Page
292
Recommended Citation
KOCZAN, LEOPOLD and ZAPRAWA, PAWEL
(2017)
"Koebe sets for the class of functions convex in two directions,"
Turkish Journal of Mathematics: Vol. 41:
No.
2, Article 6.
https://doi.org/10.3906/mat-1511-31
Available at:
https://journals.tubitak.gov.tr/math/vol41/iss2/6
