Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales

We determine the best constants C-p,C-infinity and C-1,C-p, 1 < p < infinity, for which the following holds. If u, v are orthogonal harmonic functions on a Euclidean domain such that v is differentially subordinate to u, then parallel to v parallel to(p) <= C-p,C-infinity parallel to u parallel to(infinity,) parallel to v parallel to(1) <= C-1,C-p parallel to u parallel to(p) In particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of R-2. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.