Article
Ufa Mathematical Journal
Volume 9, Number 3, pp. 148-157
Analytic functions with smooth absolute value of boundary data
Shamoyan F.A.
DOI:10.13108/2017-9-3-148
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Let $f$ be an analytic function in the unit circle $D$ continuous up to its boundary $\Gamma$, $f(z) \neq 0$, $z \in D$. Assume that on $\Gamma$, the function $f$ has a modulus of continuity $\omega(|f|,\delta)$. In the paper we establish the estimate $\omega(f,\delta) \leq A\omega(|f|, \sqrt{\delta})$, where $A$ is a some non-negative number, and we prove that this estimate is sharp. Moreover, in the paper we establish a multi-dimensional analogue of the mentioned result.
In the proof of the main theorem, an essential role is played by a theorem of Hardy-Littlewood type on H\"older classes of analytic functions in the unit circle.