A function \(f(z) = -1/z + c _0 + c_1 z + \cdots,\) analytic and nonvanishing in the punctured unit disk \(D' = \{ z: 0 0\) for all nonreal \(z \in D'\). Denote the class of all such functions by \(Q\). The authors apply the result by \textit{S.~Ruscheweyh} [Math. Z. 142, 19-3 (1975; Zbl 0292.30013)] to obtain sharp estimates of the functional \(\Re [z f'(z) / f(z)]\), \(z \in D',\) on the class \(Q\). As a consequence, the radius of starlikeness of order \(\alpha,\) \(-\infty < \alpha < 1,\) is determined in \(Q\). Finally, the intersection of the two disks \(|z \pm i |< \sqrt{2}\) is shown to be a maximal domain of univalence for the class \(Q\).