Some sufficient conditions for starlike functions associated with parabolic regions (Q1879076)

\(S_{p}\) is the well known class of functions introduced and sistematically studied by F. Rønning in 1993, by using the parabolic region, related to the class of uniformly convex functions \((UCV)\) defined by Goodman in 1991. Precisely \(f\in UCV\) if and only if \(zf^{\prime} \in S_{p}\). The classes \(S_{p}\) are generalized to the classes \(S_{p}(\alpha)\) respectively \(UCV (\alpha)\) in a natural way. The author obtains some conditions for inclusion \(S_{p}(\alpha) \subset H^{\lambda}(\alpha)\) where \(H^{\lambda}(\alpha)\) is the class of all \(\lambda\)-spirallike functions of order \(\alpha\). Also he studies the operator \(g(z)= z\left(f(z)/z \right)^{\gamma}\) and the conditions so that \(f\in S^{*}[A, B]\) implies \(g(z)\in H^{\lambda}(\alpha)\). It would have been appropiate if the author had also mentioned the conditions for the choice of any branch for the power \((f(z)/z)^{\gamma}\).