Analytic functions with smooth absolute value of boundary data

Abstract. Let f be an analytic function in the unit circle D continuous up to its boundary Γ, f(z) 6= 0, z ∈ D. Assume that on Γ, the function f has a modulus of continuity ω(|f |, δ). In the paper we establish the estimate ω(f, δ) 6 Aω(|f |, √ δ), where A is a some non-negative number, and we prove that this estimate is sharp. Moreover, in the paper we establish a multi-dimensional analogue of the mentioned result. In the proof of the main theorem, an essential role is played by a theorem of Hardy-Littlewood type on Hölder classes of the functions analytic in the unit circle.