Variation of diametrically symmetric or elliptically schlicht conformal mappings (Q1411301)

For a given \(z\) let \(z^*\) denote the antipodal point on the Riemann sphere. For a given set \(A\) let \(A^*\) be the set of all antipodal points. The set \(A\) is called diametrically symmetric if \(A= A^*\). A univalent and analytic function \(f\) on a diametrically symmetric domain \(\mathbb{D}\) is called diametrically symmetric if \(f(z^*)= f(z)^*,\) \(z\in \mathbb{D}\). Let \(\mathbb{D}\) be a diametrically symmetric domain of finite connectivity and nondegerate boundary components, and let \(S(\mathbb{D})\) be the class of univalent diametrically symmetric analytic mappings of \(\mathbb{D}\). For the class \(S(\mathbb{D})\) a variation is constructed. In its derivation a new idea of conformal sewing is used. An aplication to a (quantitative) solution of an extremal problem is given. The paper also contains several deep remarks about various variational techniques and elliptically schlicht mappings.