Counterexamples Concerning Quasiconformal Extensions of Strongly Starlike Functions

M. Fait, J. Krzyz and J. Zygmunt proved that a strongly starlike function of order α on the unit disk can be extended to a k-quasiconformal mapping with k ≤ sin(απ/2) on the whole complex plane which fixes the point at infinity. An open question is whether such a function can be extended to a k-quasiconformal mapping with k ≤ α to the whole plane In this paper we will give a negative approach to the question. This research was supported by NNSF of China (Grant No. 10231040) and NCET (06-0504)