2015/04/07 by Kalaj, David
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1504.01686
We first prove the following generalization of Schwarz lemma for harmonic mappings. Let u be a harmonic mapping of the unit ball onto itself. Then we prove the inequality ‖u(x)-(1-‖x‖2)/(1+‖x‖2)n/2 u(0)‖≤ U(|x| N). By using the Schwarz lemma for harmonic mappings we derive Heinz inequality on the boundary of the unit ball by providing a sharp constant Cn in the inequality: ‖∂r u(rη)‖r=1≥ Cn, ‖η‖=1, for every harmonic mapping of the unit ball into itself satisfying the condition u(0)=0, ‖u(η)‖=1.