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Region of variability for concave univalent functions. (English) Zbl 1176.30026

A function defined on the open unit disk \(\mathbb{D}\) is concave if the complement of the image domain \(f(\mathbb{D})\) with respect to \(\mathbb{C}\) is convex. The authors consider the class of concave functions \(f\) with opening anlge of \(f(\mathbb{D})\) at \(\infty\) less than or equal to \(\pi\alpha\) for some \(\alpha \in (0,2]\). For fixed \(z_0\in \mathbb{D}\), a given open angle and fixing \(f''(0)\) suitably, the author obtains the region of variability of \(\log f'(z_0)\) when \(f\) ranges over the class of concave univalent functions.

MSC:

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