# Remarks on some convex families of holomorphic mappings related to the convex mappings of the ball in $\mathbb{C}^n$

## Jerry Muir Jr.

created by risa on 12 Apr 2018

17 apr 2018 -- 16:00

Aula D'Antoni, Dip.Matematica, Università "Tor Vergata", Roma

Abstract.

Nice representations of various geometric families of analytic functions on the unit disk, more accurately of their closed convex hulls, are obtained by utilizing the Herglotz representation of the Carathéodory class together with some elementary analytic characterizations of the families. These representations can then be used to give simple proofs of properties of (the closed convex hulls of) the families.

As usual, things are not so clear cut in higher dimensions. Here, we present some results involving the family of normalized convex (biholomorphic) mappings of the Euclidean ball, working towards an understanding of this family that mirrors what is known in the one-dimensional setting.

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