Duality techniques for certain integral transforms to be starlike (Q1827095)

For the class \({\mathcal A}\) of normalized analytic functions in the unit disk consider the integral operator \[ V_\lambda(f)(z)= \int^1_0\lambda(t)\, f(tz)/t\,dt, \] where \(\lambda: [0,1]\to \mathbb{R}\) is a nonnegative weight function normalized by \(\int^1_0\lambda(t)\,dt= 1\). The main result of this paper are conditions such that \(V_\lambda(f)\) is starlike of order \(\mu\) \((0\leq \mu\leq 1/2)\) when \(f\) has the property \[ \text{Re}\{e^{i\varphi}[(1- \gamma) f(z)/z+ \gamma f'(z)- \beta]\}> 0, \] for \(z\) in the open unit disk, where \(0\leq \gamma\leq 1\), \(\beta< 1\) and \(\varphi\in \mathbb{R}\). Examples are given in order to show the improvement of earlier results.