Analytic functions with smooth absolute value of boundary data
Faizo Agitovich Shamoyan Β· Ufimskii Matematicheskii Zhurnal Β· 2017
Let π be an analytic function in the unit circle π· continuous up to its boundary Ξ, π (π§) ΜΈ = 0, π§ β π·.Assume that on Ξ, the function |π | has a modulus of continuity π(|π |, πΏ).In the paper we establish the estimate π(π, πΏ) π΄π(|π |, β πΏ), where π΄ 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 HΓΆlder classes of the functions analytic in the unit circle.