REMARK ON THE MAXIMUM OF THE ABSOLUTE VALUE OF AN ANALYTIC FUNCTION ON THE BOUNDARY

摘要

Let Λ~α be the analytic Hoelder class in the unit disk D. For f ∈ Λ~α and I is contained in (partial deriv)D, let M_f(I) = max_I |f|. Assume that I and J are arcs such that |J| = 2|I| and J and I have the same midpoint. Then M_f(J) ≤ C(α,f) (|I|~α + M_f(I))/(log~α(|I|~α)/(M_f(I)) + 2). It is proved that this estimate cannot be improved.