Region of variability for close-to-convex functions

For a complex number α with Re α > 0 let K φ(α) be the class of analytic functions f in the unit disk D with f (0) = 0 satisfying Re(f ′(z)/φ′(z)) > 0 in D, f ′(0)/φ′(0) = α, for some convex univalent function φ in D. For any fixed z 0 ∈ D we shall determine the region of variability for f (z 0) when f ranges over the class K φ(α). As corollaries, we present a number of its consequences.