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Regions of variability for convex functions. (English) Zbl 1105.30007

Summary: Let \({\mathcal C}\) be the class of convex univalent functions \(f\) in the unit disc \(\mathbb{D}\) normalized by \(f(0)=f'(0)-1=0\). For \(z_0\in\mathbb{D}\) and \(|\lambda|\leq 1\) we shall determine explicitly the regions of variability \(\{\log f'(z_0):f\in {\mathcal C}, f''(0)=2\lambda\}\).

MSC:

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

[1] Univalent Functions (Springer-Verlag, New York, 1983).
[2] Finkelstein, Proc. Amer. Math. Soc. 18 pp 412– (1967)
[3] Gronwall, Proc. Nat. Acad. Sci. U. S. A. 6 pp 300– (1920)
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