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- W2785463072 abstract "In this paper we are concerned with extremal problems for mappings with g-parametric representation on the unit polydisc $$mathbb {U}^2$$ of $$mathbb {C}^2$$ , where g is a univalent holomorphic function on the unit disc $$mathbb {U}$$ such that g(0) = 1, and which satisfies some natural conditions. In the first part of the paper, we obtain certain results related to extreme points and support points associated with the Carathéodory family $${mathcal M}_g(mathbb {U}^n)$$ and the family $$S_g^*(mathbb {U}^n)$$ of g-starlike mappings on $$mathbb {U}^n$$ . In particular, if g is a convex function on $$mathbb {U}$$ , we use an analogue of the shearing process due to F. Bracci, to obtain sharp coefficient bounds for the family $${mathcal M}_g(mathbb {U}^2)$$ . In the last part of the paper, we are concerned with support points for the family $$S_g^0(mathbb {U}^2)$$ of mappings with g-parametric representation on $$mathbb {U}^2$$ , where g is a convex function on $$mathbb {U}$$ with g(0) = 1, ℜg(ζ) > 0, $$zeta in mathbb {U}$$ , and which satisfies certain natural conditions. Sharp coefficient bounds for the family $$S_g^0(mathbb {U}^2)$$ , and various consequences and examples are obtained. Certain questions and conjectures are also formulated. This work complements recent work on extremal problems on the Euclidean unit ball $$mathbb {B}^2$$ in $$mathbb {C}^2$$ ." @default.
- W2785463072 created "2018-02-23" @default.
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- W2785463072 date "2017-11-08" @default.
- W2785463072 modified "2023-09-23" @default.
- W2785463072 title "Extremal Problems for Mappings with g-Parametric Representation on the Unit Polydisc in ℂ n" @default.
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- W2785463072 doi "https://doi.org/10.1007/978-3-319-70154-7_9" @default.
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