Computational Methods and Function Theory 7 (2007), No. 2, 573--582
Copyright Heldermann Verlag 2007
Martin Lamprecht
martin@ucy.ac.cy , The University of Cyprus, Department of Mathematics and Statistics, P.O. Box 20537, 1678 Nicosia, Cyprus.
We will show that the set of starlike univalent functions in $\mathbb{D}$ is starlike in the Hornich space, i.e.\ for starlike functions $f$ and $0\leq \alpha\leq 1$, the function $\int_0^z (f'(\zeta))^\alpha \,d\zeta$ is also starlike. This solves a problem given by Kim, Ponnusamy and Sugawa in \cite{kimposu}. An important step in proving this result will be to show that for starlike functions $f$ and $z\in\mathbb{D}$ we have ${|\int_0^1 \arg(z/\gamma'(t))\,dt| < \pi/2}$, where $\gamma(t):=f^{-1}(tf(z))$, $0\leq t \leq 1$.
Keywords: Starlike functions; Hornich operations; Integral transform.
MSC 2000: 30C45; 30C80; 44A05.
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