Variations of univalent functions due to holomorphic motions (Q1899783)

Let \(K = \{\lambda \in\mathbb{C} : |\lambda |< 1\}\) and let \(E \subset\mathbb{C}\) be a subset of the complex plane \(\mathbb{C}\) with more than three points. A function \(M : K \times E \to\mathbb{C}\) is a motion of \(E\) with parameter \(\lambda\) if (i) \(M(0,z) = z\) for each \(z \in E\), (ii) \(M (\lambda \cdot) : E \to\mathbb{C}\) is an injective function for each \(\lambda \in K\). The motion \(M\) with parameter \(\lambda\), is called continuous (differentiable holomorphic, etc.) if \(M (\cdot, z) : K \to\mathbb{C}\) is continuous (differentiable, holomorphic, etc.) for each \(z \in E\). Let \(G \subset\mathbb{C}\) be an open and simply connected set and let \(a \in G\). The Riemann mapping of \(G\) normalized in \(a\) is the unique function \(R(G,a) : K \to G\) holomorphic with \(R(G,a) (0) = a\) and \(R'(G,a) (0) > 0\). At last, let \(\Psi_r : K \to K\) be the holomorphic quasiconformal solution of the Beltrami differential equation \(\overline \partial \Psi_r/ \partial \Psi_r = r\) normalized by \(\Psi_r (0) = 0\), \(\Psi_r (1) = 1\), where \(r:\mathbb{C}\to \mathbb{C}\) is a measurable function with \(|r |_\infty < (K - 1)/(K + 1)\), \(K \geq 1\). The author 1) studies variations of univalent (injective) holomorphic functions which are obtained by holomorphically parametrized motions, 2) concentrates on quasiconformal motions of \(E\) with a holomorphic or real analytic parameter, 3) obtains an integral representation for the quasiconformal mappings \(\psi_r\), 4) calculates the variations of \(R(G, a)\), 5) gives some examples of variations of univalent functions.