Smoothness of a Holomorphic Function and Its Modulus on the Boundary of a Polydisk

Abstract

We prove that if a function f is holomorphic in the polydisk ????ⁿ, n ≥ 2, f is continuous in \( \overline{{\mathbb{D}}^n} \), f(z) ≠ 0, z ∈ ????ⁿ, and |f| belongs to the α-Hölder class, 0 < α < 1, on the boundary of ????ⁿ, then f belongs to the \( \left(\frac{\alpha }{2}-\varepsilon \right) \)-Hölder class on \( \overline{{\mathbb{D}}^n} \) for any ε > 0.