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Radius constants for functions with the prescribed coefficient bounds. (English) Zbl 1470.30004

Summary: For an analytic univalent function \(f(z) = z + \sum_{n = 2}^{\infty} a_n z^n\) in the unit disk, it is well-known that \(\left|a_n\right| \leq n\) for \(n \geq 2\). But the inequality \(\left|a_n\right| \leq n\) does not imply the univalence of \(f\). This motivated several authors to determine various radii constants associated with the analytic functions having prescribed coefficient bounds. In this paper, a survey of the related work is presented for analytic and harmonic mappings. In addition, we establish a coefficient inequality for sense-preserving harmonic functions to compute the bounds for the radius of univalence, radius of full starlikeness/convexity of order \(\alpha\) (\(0 \leq \alpha < 1\)) for functions with prescribed coefficient bound on the analytic part.

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
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