On a form of the direct variational method for univalent functions (Q2387107)

For a given continuum \(\Gamma\) which complement is a simple connected domain we define a deleted neighbourhood \(U(\Gamma)\) as a double connected domain such that one component is a continuum \(\Gamma\) and the second component contains infinity. In this paper some scheme to construction the variational methods is given. In particular the following result is proved: Theorem 1: Let \(F\) be a nondegenerate continuum whose complement consists of a simply connected domain. Let \(\zeta(\xi)\) be a conformal mapping of a certain \(U (\Gamma)\) on some appropriate sing \(D(\delta,1)=\{\zeta<|\zeta|<1\}\) such that \(\Gamma\) corresponds to the internal boundary component of \(D(\delta,1)\). We assume that in \(U(\Gamma)\) the following relation holds: \[ p(\xi)=\frac{i\zeta(\xi)} {\zeta'(\xi)}Q \bigl(\zeta (\xi)\bigr), \] where \(Q(\zeta)\) is regular in \(D(\delta,1)\) and has a nonnegative real part on \(|\zeta|=\delta\). Then there exists, in a complement of the continuum \(\Gamma\), the univalent variation \(V(w,\tau)\) such that on any closed subset \(E\) of the complement of \(\Gamma\) and for sufficiently small values \(\tau\), \(0<\tau<\tau^*\), the uniform estimate \[ V(w,\tau)=w+\tau \int_\gamma \frac{p(\xi)}{\xi-w}d\xi+o(\tau) \] is valid. Here \(\gamma\) is a simple closed rectifiable path which seperates the boundary components of this neighbourhood \(U(\Gamma)\) and detaches the set \(E\) from the continuum \(\Gamma\) and \(o(\tau)\) stands for a function of variables \(w,\tau\) such that \(o(\tau)/ \tau\) tends uniformly to 0 on \(E\) if \(\tau\to 0^+\).