Holomorphic motions and quasiconformal extensions (Q2769214)

Let \(f\) be an analytic function defined in the unit disk with \(f(0)= 0\), \(f'(0)\neq 0\) and let \(p(z)\) denote one of the following three differential expressions NEWLINE\[NEWLINEf'(z),\quad zf'(z)/f(z),\quad 1+ z''(z)/f'(z).\tag{\(*\)}NEWLINE\]NEWLINE If, for \(0\leq k< 1\), \(p(z)\) satisfies the condition NEWLINE\[NEWLINE|p(z)- (1+ k^2)/(1- k^2)|\leq 2k/(1- k^2),\quad|z|< 1NEWLINE\]NEWLINE then it is shown as the main result that \(f\) can be extended to a \(k\)-quasiconformal mapping of the whole plane. By considering the example \(p(z)= (1+ kz^2)/(1- kz^2)\) and the corresponding (normalized) univalent function \(f\) it is shown in addition that, in the first two cases of \((*)\), the result is best possible which means that the quasiconformal extension is extremal. In order to prove the result above and other similar but more general results on functions that e.g. are spirallike or close-to-convex the author constructs suitable holomorphic motions and then applies the \(\lambda\)-lemma.