Quasiconformal extension of strongly spirallike functions

TL;DR

This paper demonstrates that strongly $ ext{lambda}$-spirallike functions of order $ ext{alpha}$ can be extended to quasiconformal automorphisms of the complex plane, providing geometric characterizations and explicit mapping functions.

Contribution

It introduces a method to extend strongly $ ext{lambda}$-spirallike functions to quasiconformal automorphisms and offers geometric characterizations and explicit forms of these functions.

Findings

Extension to $ ext{sin}(rac{ ext{pi} ext{alpha}}{2})$-quasiconformal automorphisms

Geometric characterizations of strongly $ ext{lambda}$-spirallike domains

Explicit form of the standard strongly $ ext{lambda}$-spirallike mapping function

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.