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 $\ID=\{z :|z|<1\}$ 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 $z_0 \in \ID$ and $f"(0)$ whenever $f$ varies over the class $\mathcal{C}(A,B)$.

9 pages, To appear in Proc. Indian Acad. Sci. Math. Sci