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.

10 pages

References in corpus (1)

• Correspondence between spirallike functions and starlike functions