For Kβ₯1, suppose that f is a K-quasiconformal self-mapping of the
unit ball Bn, which satisfies the following: (1) the
polyharmonic equation Ξmf=Ξ(Ξmβ1f)=Οmβ(ΟmββC(Bn,Rn)), (2)
the boundary conditions
Ξmβ1fβ£Snβ1β=Οmβ1β,Β β¦,Β Ξ1fβ£Snβ1β=Ο1β
(ΟkββC(Snβ1,Rn) for
jβ{1,β¦,mβ1} and Snβ1 denotes the unit sphere in
Rn), and (3)f(0)=0, where nβ₯3 and mβ₯2 are
integers. We first establish a Heinz type inequality on mappings satisfying the
polyharmonic equation. Then we use the obtained results to show that f is
Lipschitz continuous, and the estimate is asymptotically sharp as Kβ1 and
β₯Οjββ₯βββ0 for jβ{1,β¦,m}.Comment: 28 page