Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales

Abstract

AbstractWe determine the best constants Cp,∞ and C1,p, 1 < p < ∞, for which the following holds. If u, v are orthogonal harmonic functions on a Euclidean domain such that v is differentially subordinate to u, thenIn particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of ℝ2. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.