Variability regions of close-to-convex functions

Abstract

M. Biernacki gave concrete forms of the variability regions of $z/f(z)$ and $zf'(z)/f(z)$ of close-to-convex functions $f$ for a fixed $z$ with $|z|<1$ in 1936. The forms are, however, not necessarily convenient to determine the shape of the full variability region of $zf'(z)/f(z)$ over all close-to-convex functions $f$ and all points $z$ with $|z|<1.$ We will propose a couple of other forms of the variability regions and see that the full variability region of $zf'(z)/f(z)$ is indeed the complex plane minus the origin. We also apply them to study the variability regions of $\log[z/f(z)]$ and $\log[zf'(z)/f(z)].$