Smoothness of a Holomorphic Function in a Ball and of its Modulus on the Sphere

Abstract

Let a function f be holomorphic in the unit ball ????ⁿ, continuous in the closed ball \( {\overline{\mathbb{B}}}^n \) , and let f(z) ≠ 0, z ∈ ????ⁿ. Assume that |f| belongs to the α-Hölder class on the unit sphere Sⁿ, 0 < α ≤ 1. The present paper is devoted to the proof of the statement that f belongs to the α/2-Hölder class on \( {\overline{\mathbb{B}}}^n \).