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Regions of variability for univalent functions. (English) Zbl 0598.30028

Denote by S the customary class of schlicht functions f, and by \(S_ 0\) the class of univalent functions g in \({\mathbb{D}}\) such that g(z)\(\neq 0\) and \(g(0)=1\). The authors compare the regions of variability \(\tilde W(r)=\{(1-r^ 2)f'(r):\quad f\in S\}\) and \(W_ 0(r)=\{g(r):\quad g\in S_ 0\},\) \(0<r<1\), and give a list of indications in support of the conjecture that \(W_ 0(r)=\tilde W(r)\). In particular, they prove the inclusion \(W_ 0(r)\subset \tilde W(r)\) and numerical computation shows that \(W_ 0(r)\) visibly fills \(\tilde W(\)r). As a surprise however, they prove that \(W_ 0(r)\) and \(\tilde W(\)r) are distinct.
Reviewer: A.Pfluger

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
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