Analytic functions with smooth absolute value of boundary data

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 Hölder classes of the functions analytic in the unit circle.