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arXiv · 1705.05738

On Becker's univalence criterion

Juha-Matti Huusko·Toni Vesikko
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Abstract

We study locally univalent functions $f$ analytic in the unit disc $\mathbb{D}$ of the complex plane such that $|{f"(z)/f'(z)}|(1-|z|^2)\leq 1+C(1-|z|)$ holds for all $z\in\mathbb{D}$, for some $0<C<\infty$. If $C\leq 1$, then $f$ is univalent by Becker's univalence criterion. We discover that for $1<C<\infty$ the function $f$ remains to be univalent in certain horodiscs. Sufficient conditions which imply that $f$ is bounded, belongs to the Bloch space or belongs to the class of normal functions, are discussed. Moreover, we consider generalizations for locally univalent harmonic functions.

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Juha-Matti Huusko, Toni Vesikko. 2017-05-16. On Becker's univalence criterion. https://arxiv.org/abs/1705.05738

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