Holomorphic motions and quasicircles (Q2846917)

The author establishes new dilatation estimates for images of circles under conformal mappings using holomorphic motions. Such images of circles are usually \(K\)-quasicircles and there are few sharp estimates on \(K\) in these settings. The author proves such estimates and discusses their sharpness or how far they are from being sharp. He also gives a proof for the best possible estimate for \(K\) in the case of the ellipse.NEWLINENEWLINEThe first main result, Theorem 2, states that if \(\varphi:\mathbb D @>>> \mathbb C\) is a conformal mapping, \(0<r<1\), then \(\varphi( \mathbb S (r))\) is a \(\sqrt{\frac{1+r}{1-r}}\)-quasicircle where \(\mathbb S (r)=\{|z|=r\}\). Then, a particular example using the Koebe function is compared with Theorem 2. Theorem 3 gives a dilatation estimate for \(\varphi( \mathbb S (r))\) when \(\varphi\) is a conformal mapping from the annulus \(\mathbb A\left(r_0,1\right)=\left\{r_0<|z|<1\right\}\), \(r_0<r<1\), with \(\varphi(\mathbb S)=\mathbb S\) where \(\mathbb S =\mathbb S(1)\).NEWLINENEWLINEDetailed discussions and several references also help the reader.