Quasiconformal extension of strongly spirallike functions

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

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