LOCAL SMOOTHNESS OF AN ANALYTIC FUNCTION COMPARED TO THE SMOOTHNESS OF ITS MODULUS

摘要

Let Φ be a function analytic in the disk and continuous up to the boundary, and let its modulus of continuity satisfy the H?lder condition of order α, 0 < α < 2, at a single boundary point. Under standard assumptions on the zeros of Φ, this function must be then at least α/2-H?lder (in a certain integral sense) at the same point. There are generalizations to not necessarily power-type H?lder smoothness.