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**On uniformly starlike functions.**
*(English)*
Zbl 0726.30013

The author introduces a new class of functions which are uniformly starlike in the unit disc. This class seems to be interesting with respect to the geometrical theory of univalent functions. A function f(z) is said to be uniformly starlike in E: \(| z| <1\) if f is starlike and has the property that, for every circular arc \(\gamma\) contained in E, with center \(\xi\) also in E, the arc f(\(\gamma\)) is starlike with respect to f(\(\xi\)). (This class is denoted by UST.) The author proves the basic properties of the class UST. Among other things, he shows that the answer to the following Pinchuk question is negative. The Pinchuk question is: If f(z) is starlike, is it true that f(\(\gamma\)) is a closed curve that is starlike with respect to f(\(\xi\)), where \(\gamma\) is a circle contained in E with center \(\xi\).

Reviewer: S.Walczak (Łódź)

### MSC:

30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |

30C50 | Coefficient problems for univalent and multivalent functions of one complex variable |

30C20 | Conformal mappings of special domains |

### Keywords:

uniformly starlike functions
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\textit{A. W. Goodman}, J. Math. Anal. Appl. 155, No. 2, 364--370 (1991; Zbl 0726.30013)

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### References:

[1] | Brown, J. E., Images of disks under convex and starlike functions, Math. Z., 202, 457-462 (1989) · Zbl 0662.30008 |

[2] | Goodman, A. W., Univalent functions and nonanalytic curves, (Proc. Amer. Math. Soc., 8 (1957)), 598-601 · Zbl 0166.33002 |

[3] | Goodman, A. W., Univalent Functions (1983), Polygonal: Polygonal Washington, NJ · Zbl 0166.33002 |

[4] | Obradovic, M., Some theorems on subordination by univalent functions, Mat. Vesnik, 37, 211-214 (1985) · Zbl 0591.30010 |

[5] | Pommerenke, C., Linear-invariante Familien analytischer Funktionen, I, Math. Ann., 155, 108-154 (1964) · Zbl 0128.30105 |

[6] | Rudin, W., Function Theory in Polydiscs (1969), Benjamin: Benjamin New York · Zbl 0177.34101 |

[7] | Sakaguchi, K., On a certain univalent mapping, J. Math. Soc. Japan, 11, 72-75 (1959) · Zbl 0085.29602 |

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