Functions in Bloch-type spaces and their moduli

Abstract

Given a suitably regular nonnegative function $\omega$ on $(0,1]$, let ${\mathcal{B}}_{\omega }$ denote the space of all holomorphic functions $f$ on the unit ball ${\mathbb{B}}_n$ of ${\mathbb{C}}^n$ that satisfy $$|\nabla f(z)|\le C\frac{\omega (1-|z|)}{1-|z|},z\in {\mathbb{B}}_n,$$ with some fixed $C=C_f>0$. We obtain a new characterization of ${\mathcal{B}}_{\omega }$ functions in terms of their moduli.

9 pages; to appear in Ann. Acad. Sci. Fenn. Math. 41 (2016), No. 2

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