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Coefficient inequalities and maximalization of some functionals for pairs of vector functions. (English) Zbl 0547.30015

The mappings F and G, \(F=(F_ 1,F_ 2,...,F_ m):\Delta\to {\mathbb{C}}^ m\) when \(m\geq 1\), \(G=(G_ 1,...,G_ n):\Delta\to {\mathbb{C}}^ n\) when \(n\geq 1\), \(F=0\) or \(G=0\) when \(m=0\) or \(n=0\), respectively, are said to form a pair (F,G) if \(F_ 1,...,F_ m\) and \(G_ 1,...,G_ n\) are functions analytic and univalent in \(\Delta\) (the unit disc) and satisfy: \(F_ k(z)\neq F_ j(\chi),\quad k\neq j,\quad n\geq 0,\quad G_ k(z)\neq G_ j(\chi),\quad k\neq j\quad F_ k(z)G_ j(\chi)\neq 1\) for z,\(\chi \in\Delta \). The author finds many results for this general class and points out that various classes are particular cases of the above.
(Among them: Bieberbach-Eilenberg class, Gelfer’s functions and so on.)
Reviewer: D.Aharonov

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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