Quasiconformal distortion of univalent functions via holomorphic motions (Q1128252)

Let \(\Delta\{\zeta\in\mathbb{C}:| \zeta|<1\}\) and let \(h: \Delta\to h(\Delta)\) be a univalent function. Let \(\mu(\lambda):\Delta\to \Delta\) be a measurable function with \(|\mu(\lambda)|_\infty< 1\), depending holomorphically on \(\lambda\) in a neighbourhood \(V\) of \(0\in\mathbb{C}\) as element of the Lebesgue space \(L^\infty(\Delta)\), and \(\mu(0)= 0\), that is \[ \mu(\lambda)(\zeta)= \mu_1(\zeta)\lambda+ \mu_2(\zeta)\lambda^2+\cdots\;. \] Let \(v(\lambda):\mathbb{C}\to\Delta\) is defined by \[ v(\lambda)(z)= \begin{cases} (\mu(\lambda)){h'\over\overline{h'}}\circ h^{-1}(z)\quad &\text{if}\quad z\in h(\Delta)\\ 0\quad &\text{otherwise}.\end{cases} \] Then \(v(\lambda)\) depends holomorphically on \(\lambda\) in \(V\) as an element of \(L^\infty(\mathbb{C})\) with \(| v(\lambda)|_\infty< 1\). Let at last \(g_\lambda:\Delta\to \Delta\) and \(f_\lambda:\overline{\mathbb{C}}\to\overline{\mathbb{C}}\) be the homeomorphic quasiconformal solutions of the Beltrami differential equations \[ \mu(\lambda)= {\overline\partial g_\lambda\over \partial g_\lambda}\quad\text{and} \quad v(\lambda)= {\overline\partial f_\lambda\over\partial f_\lambda} \] normalized by \(g_\lambda(0)= 0\), \(g_\lambda(1)= 1\), \(f_\lambda(0)= 0\), \(f_\lambda(1)= 1\), \(f_\lambda(\infty)= \infty\). One defines \(H:V\times\Delta\to \mathbb{C}\), \(H(\lambda,\zeta):= h_\lambda(\zeta)= (f_\lambda\circ h\circ g^{-1}_\lambda)(\zeta)\). It appeared that \(h_\lambda\) is univalent on \(\Delta\) for all \(\lambda\in V\). The author proves that \(H\) depends real analytically on \(\text{Re }\lambda\), \(\text{Im }\lambda\), and calculates the variations of \(h_\lambda\).