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)].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap111-1-7,
author = {Takao Kato and Toshiyuki Sugawa and Li-Mei Wang},
title = {Variability regions of close-to-convex functions},
journal = {Annales Polonici Mathematici},
volume = {111},
year = {2014},
pages = {89-105},
zbl = {1302.30015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap111-1-7}
}
Takao Kato; Toshiyuki Sugawa; Li-Mei Wang. Variability regions of close-to-convex functions. Annales Polonici Mathematici, Tome 111 (2014) pp. 89-105. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap111-1-7/