Subordinators by alpha-convex functions (Q1876291)

Let \(A\) denote the class of functions \(\phi\) univalent and holomorphic on the unit disk \(U\) with \(\phi(0)=0\) and \(\phi(z)\neq 0,\;\forall z\in U\). For fixed \(\phi,\varphi\in A\) and \(\beta, \gamma\neq 0\), the author determines conditions on the univalent function \(f\) so that the subordination \[ \left [ \frac{\phi(z)}{\varphi(z)+(1/\gamma)z\varphi^\prime(z)} \right ]^{1/\beta}f(z)\prec k(z) \] implies the subordination \[ \left [ \frac{\gamma}{z^\gamma\varphi(z)}\int_0^zf^\beta(t)t^{\gamma-1}\phi(t)\,dt \right ]^{1/\beta} \prec k(z) \] for all \(1/\beta\)-convex univalent functions \(k\).