Smoothness of a holomorphic function and its modulus on the boundary of a polydisk

We prove that if a function $f$ is holomorphic in the polydisk $\mathbb D^n$, $n\ge2$, $f$ is continuous in $\overline{\mathbb D^n}$, $f(z)\ne0$, $z\in\mathbb D^n$, and $|f|$ belongs to the $\alpha$-Hölder class, $0<\alpha<1$, on the boundary of $\mathbb D^n$ then $f$ belongs to the $(\frac\alpha2-\varepsilon)$-Hölder class on $\overline{\mathbb D^n}$ for any $\varepsilon>0$.