Variability regions of close-to-convex functions

Volume 111 / 2014

Takao Kato, Toshiyuki Sugawa, Li-Mei Wang Annales Polonici Mathematici 111 (2014), 89-105 MSC: Primary 30C45; Secondary 30C75. DOI: 10.4064/ap111-1-7


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


  • Takao KatoGraduate School of Science and Engineering
    Yamaguchi University
    Yoshida, Yamaguchi 753-8512, Japan
  • Toshiyuki SugawaGraduate School of Information Sciences
    Tohoku University
    Aoba-ku, Sendai 980-8579, Japan
  • Li-Mei WangSchool of Statistics
    University of International Business and Economics
    No. 10, Huixin Dongjie, Chaoyang District
    Beijing 100009, China

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