On C∞-Compactness of Quasiconformal Harmonic Maps on the Poincare Disk

Let {f(n) : D -> D} be a sequence of K-quasiconformal harmonic maps on the unit disk D with respect to the Poincare metric. It is known that there is a subsequence of {f(n)}that uniformly converges on (D) over bar and the limit function is either a K-quasiconformal harmonic map of the Poincare disk or a constant. In this paper, it is shown that, if the limit function is not a constant, the subsequence can be chosen such that its derivative sequence of arbitrary order uniformly converges to the corresponding derivative of the limit function on compact subsets of D.