Schwarz Lemma for Harmonic Mappings in the Unit Ball

摘要

If 1 = p = infinity and u is a harmonic mapping of the unit ball B-n into itself such that u(0) = 0, then we obtain the sharp inequality vertical bar u(x)vertical bar = g(p)(vertical bar x vertical bar)parallel to u parallel to(p) for some smooth function g(p) vanishing in 0, where parallel to u parallel to(infinity) := sup(x is an element of Bn) vertical bar u(x)vertical bar for p = +infinity, and otherwise .