Regions of variability for a class of analytic and locally univalent functions defined by subordination

Abstract

In this article, we consider a family $\mathcal {C}(A, B)$ of analytic and locally univalent functions on the open unit disc $\mathbb {D}=\{z :|z|<1\}$ in the complex plane that properly contains the well-known Janowski class of convex univalent functions. In this article, we determine the exact set of variability of $\log (f^{\prime }(z_{0}))$ with fixed $z_{0} \in \mathbb {D}$ and $f^{\prime \prime }(0)$ whenever f varies over the class $\mathcal {C}(A, B)$.