Hyperbolic metrics, homogeneous holomorphic functionals and Zalcman's conjecture

Details Hyperbolic metrics, homogeneous holomorphic functionals and Zalcman's conjecture Hyperbolic metrics, homogeneous holomorphic functionals and Zalcman's conjecture

2011-09-21

arxiv.org/abs/1109.4646v1

Mathematics

Complex Variables

This is a revised and improved version of arXiv:0907.3623. Comments are welcome

Scientific paper

We show, using the Kobayashi and Caratheodory metrics of the universal Teichmuller space, that a wide class of holomorphic functionals on the space of univalent functions in the disk is maximized by the Koebe function or by its root transforms; their extremality is forced by hyperbolic features. As consequences, this implies the proofs of the well-known Zalcman and Bieberbach conjectures.

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