Smoothness of a holomorphic function in a ball and smoothness of its modulus on the sphere

Let a function $f$ be holomorphic in the unit ball $\mathbb B^n$, continuous in the closed ball $\overline{\mathbb B}^n$, and let $f(z)\ne0$, $z\in\mathbb B^n$. Assume that $|f|$ belongs to the $\alpha$-Hölder class on the unit sphere $S^n$, $0<\alpha\leq1$. The present paper is devoted to the proof of statement that $f$ belongs to the $\alpha/2$-Hölder class on $\overline{\mathbb B}^n$.