An extension of Jenkin's condition for extremal domains associated with the univalent Bloch-Landau constant
The univalent Bloch-Landau constant \(\mathcal U\) is the largest number such that the image of the unit disk under any conformal map \(f\) must contain some disk of radius \({\mathcal U} | f'(0)| .\) A simple \(C^{1}\) curve \(\gamma\) is said to be an internal boundary curve for a domain \(D\) if \(\gamma\) is a part of the boundary of \(D\) and if, to each non-endpoint \(P\) of \(\gamma\), there corresponds a positive \(\varepsilon\) for which \(D(P, \varepsilon) \backslash \gamma\) is a part of \(D\). A planar domain \(D\) that contains \(z_{0}\) and such that \(\gamma\) is an internal boundary curve for \(D\) is harmonically symmetric in \(\gamma\) with respect to \(z_{0}\) if the left and the right sides of any subarc of \(\gamma\) have the same harmonic measure at \(z_{0}.\) The author under review gives an extension of \textit{J. A. Jenkins}' [Kodai Math. J. 15, No. 1, 79--81 (1992; Zbl 0764.30018)] condition for extremal domains. The main theorems are the following: Theorem 1. Suppose that \(D\) is an extremal domain for the univalent Bloch-Landau constant. Suppose that \(\gamma\) is an internal boundary curve for \(D\), no point of which lies on the boundary of an extremal disk. Then \(D\) is harmonically symmetric in \(\gamma\) with respect to 0. Theorem 2. Suppose that the straight line segment \(\gamma\) is an internal boundary curve for a simply connected domain \(D\). Suppose further that \(D\) is harmonically symmetric in \(\gamma\) with respect to 0. Then \(\gamma\) lies on a ray emanating from the origin and \(D\) is symmetric under reflection in this ray.
scientific article; zbMATH DE number 5305529