Variability regions of close-to-convex functions
Volume 111 / 2014
Annales Polonici Mathematici 111 (2014), 89-105
MSC: Primary 30C45; Secondary 30C75.
DOI: 10.4064/ap111-1-7
Abstract
M. Biernacki gave in 1936 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 $|z|<1$. 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 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 $\log[zf'(z)/f(z)].$
