Some properties of prestarlike and universally prestarlike functions (Q1038549)

Let \({\mathcal R}_{\alpha}^{u}\) denote the classes of normalized universally prestarlike functions of order \(\alpha \leq 1\) in the slit domain \({\mathbb C} \setminus [1, \infty]\). These classes were recently introduced by \textit{S. Ruscheweyh}, \textit{L. Salinas} and \textit{T. Sugawa} [Isr. J. Math. 171, 285--304 (2009; Zbl 1203.30021)] and further investigated by the first two authors in [Math. Z. 263, No. 3, 607--617 (2009; Zbl 1181.30011)]. In the present paper, it is shown that, except for certain M\"bius transformations, there are no rational functions in \({\mathcal R}_{\alpha}^{u}\), \(\alpha < 1\). A consequence of this is that there are no numbers \(t = t_{u}(\alpha) > 0\) such that \(f \in {\mathcal R}_{1}^{u}\) implies that \(f(tz)/t \in {\mathcal R}_{\alpha}^{u}\). This differs from the situation for classical prestarlike functions in the unit disk, where such numbers \(t(\alpha) > 0\) exist and are determined by the author in this paper.