Geodesic
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

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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

Takao Kato 1 ; Toshiyuki Sugawa 2 ; Li-Mei Wang 3

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 
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Takao Kato; Toshiyuki Sugawa; Li-Mei Wang. Variability regions of close-to-convex functions. Annales Polonici Mathematici, Tome 111 (2014) no. 1, pp. 89-105. doi: 10.4064/ap111-1-7

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