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