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Locally univalent functions, VMOA and the Dirichlet space. (English) Zbl 1276.30026

Let \(\mathcal H(\mathbb D)\) be the algebra of analytic functions in the unit disk \(\mathbb D\) in the complex plane and let \(\mathbb T\) be the unit circle. The authors describe some subspaces of \(\mathcal H(\mathbb D)\) using integrability conditions or some other special behavior of \(f(z)\) or \(f'(z)\) near \(\mathbb T\): Hardy spaces consist of functions which in particular have a radial limit almost everywhere on \(\mathbb T\), \(\mathrm{BMOA}\) is the space of functions with the bounded oscillation on \(\mathbb T\), \(\mathrm{VMOA}\) is a subspace of \(\mathrm{BMOA}\), \(B\) is the Bloch space, \(B_0\) is the little Bloch space and \(D\) is the Dirichlet space.
The authors prove a series of implication results of the type:
If \(g \in \mathcal H(\mathbb D)\) is locally univalent and if \(\log g'\) belongs to a particular one of the above spaces then \(g\) or \(g'\) belongs to another of these spaces or has some other interesting property.
Many examples are given.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C55 General theory of univalent and multivalent functions of one complex variable
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