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On the univalence of an implicit function. (English) Zbl 0571.30023

If a function f in the familiar schlicht class S omits two values of equal modulus, then L. Brickman [Bull. Am. Math. Soc. 76, 372-374 (1970; Zbl 0189.088)] showed how to decompose f into a convex combination of other functions in S. In other words, such functions are not extreme points of S. The present author casts the same decomposition in a new light. In addition, using a similar construction he shows that each extreme point f for the class of Guelfer functions has the property that the range of \(f^ 2\) contains the interval (0,1].
Finally, in a variant of the same idea he establishes the following result for functions f that are analytic and univalent in a disk and bounded by a constant \(M<1\). If g is defined by \(f=g+ag^ n\) for some constant \(a\in {\mathbb{C}}\), then all branches of g are regular and univalent in this disk for all \(n\geq N\), where N depends only on M and a.
Reviewer: G.Schober

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable

Citations:

Zbl 0189.088
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