Region of variability for close-to-convex functions. II. (Q734840)

For a complex number \(\alpha\) with \(\Re \alpha>0\), let \(K_\phi(\alpha)\) be the class of analytic functions \(f\) in the unit disk \(\mathbb{D}\) with \(f (0)=0\) satisfying \(\Re( f'(z)/\phi'(z))>0\) in \(\mathbb{D}\), \(f'(0)/\phi'(0)=\alpha\), for some convex univalent function \(\phi\) in \(\mathbb{D}\). For any fixed \(z_0\in \mathbb{D}\) and \(\lambda \in \overline{\mathbb{D}}\), the authors determine the region of variability for \(f (z_0)\) when \(f\) ranges over the class \(K_\phi(\alpha,\lambda)\) consisting of functions \(f\in K_\phi(\alpha)\) satisfying the condition \((d/dz)(f'(z)/\phi'(z)) |_{z=0} = 2\lambda (\Re \alpha)\).