### 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)].

Original language | English |
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Pages (from-to) | 89-105 |

Number of pages | 17 |

Journal | Annales Polonici Mathematici |

Volume | 111 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

### Keywords

- Close-to-convex function
- Linearly accessible
- Variability region

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Kato, T., Sugawa, T., & Wang, L. M. (2014). Variability regions of close-to-convex functions.

*Annales Polonici Mathematici*,*111*(1), 89-105. https://doi.org/10.4064/ap111-1-7