One pointed univalent logharmonic mappings (Q1815448)

Let \(\Omega\) and \(D\) be domains in the complex plane \(\mathbb{C}\). Then \(\Omega\) is said to be harmonically accessible from \(D\) if there exists a univalent harmonic mapping from \(D\) onto \(\Omega\). If, in addition, \(D\) is harmonically accessible from \(\Omega\) then the domains are harmonically equivalent. A punctured disk is harmonically accessible from an annulus, but they are not harmonically equivalent. More generally, any domain of \(\mathbb{C}\) containing the point infinity can be mapped harmonically and univalently onto a pointed plane \(\Omega=\mathbb{C}\backslash\bigcup_{j\in J}\{w_j\}\), \(J\subset\mathbb{N}\), although the points \(w_j\) cannot be prescribed in general. The main result of the paper shows that this can be done in the following case. Suppose that \(K\) is a compact continuum of \(\mathbb{C}\) such that \(\mathbb{C}\backslash K\) is connected and such that \(z\in K\) implies that \(z+2\pi in\not\in K\) for all nonzero integers \(n\). Let \(K_n=\{z=\zeta+2\pi in; \zeta\in K\}\), for all integers \(n\) and let \(D^*=\mathbb{C}\backslash\bigcup K_n\). Then \(\Omega^*=\mathbb{C}\backslash\bigcup\{2\pi in\}\) is harmonically accessible from \(D^*\). This is proved by way of the authors' establishing a complete characterization of univalent logharmonic mappings from the exterior of the unit disk onto the punctured plane.