Variability regions of close-to-convex functions

Takao Kato¹ ; Toshiyuki Sugawa² ; Li-Mei Wang³

¹ Graduate School of Science and Engineering Yamaguchi University Yoshida, Yamaguchi 753-8512, Japan
² Graduate School of Information Sciences Tohoku University Aoba-ku, Sendai 980-8579, Japan
³ School of Statistics University of International Business and Economics No. 10, Huixin Dongjie, Chaoyang District Beijing 100009, China

Annales Polonici Mathematici, Tome 111 (2014) no. 1, pp. 89-105.

Résumé

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 $\log[zf'(z)/f(z)].$

Détail

DOI : 10.4064/ap111-1-7

Keywords: biernacki gave concrete forms variability regions close to convex functions fixed forms however necessarily convenient determine shape full variability region close to convex functions points propose couple other forms variability regions see full variability region indeed complex plane minus origin apply study variability regions log log