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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 \(\xi\).

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
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