A property of Koebe functions (Q1086378)

For a fixed real \(\delta\) define \(K_{\delta}(z)=(1+z)^{\delta}(1- z)^{-3\delta}\) for z in the unit disk \(\Delta\). It is proved that \(K_{\delta}(z)\) is univalent in \(\Delta\) provided \(0<\delta \leq 2/3\). Moreover, \(K_{\delta}(\Delta)\) is (i) starshaped, if \(0<\delta \leq 1/3\), (ii) convex in the direction of the real axis, if \(1/3<\delta \leq 2/3\). Both constants are best possible. This is a generalization of an earlier result of \textit{P. L. Duren} and \textit{R. McLaughlin} [Mich. Math. J. 19, 267-273 (1972; Zbl 0228.30007)].