Region of variability of univalent functions f(z) for which zf '(z) is spirallike

Abstract

Abstract. For a complex number α with Reα> 0 let Sα(λ) be the class of analytic functions f in the unit disk D with f(0) = 0 = f ′(0)−1, f ′′(0) = 2λe−iα cosα satisfying Re eiα 1 + zf ′′(z) f ′(z)> 0 for z ∈ D. For z0 ∈ D fixed, we determine the region of variability for log f ′(z0) when f ranges over the class Sα(λ). As a consequence, we obtain an estimate for a pre-Schwarzian norm for Sα(0).