Let Λ^α be the analytic Holder class in the unit disk double-struck D sign. For f ε Λ^α and I ⊂ ∂double-struck D sign, let M_f(I) = max_I |f|. Assume that I and J are arcs such that |J| = |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.