Quasiconformal extension of strongly spirallike functions

Abstract

We show that a strongly $\lambda$-spirallike function of order $\alpha$ can be extended to a $\sin(\pi\alpha/2)$-quasiconformal automorphism of the complex plane for $-\pi/2<\lambda<\pi/2$ and $0<\alpha<1$ with $|\lambda|<\pi\alpha/2.$ In order to prove it, we provide several geometric characterizations of a strongly $\lambda$-spirallike domain of order $\alpha.$ We also give a concrete form of the mapping function of the standard strongly $\lambda$-spirallike domain $U_{\lambda,\alpha}$ of order $\alpha.$ A key tool of the present study is the notion of $\lambda$-argument, which was developed by Y. C. Kim and the author.