Moduli of rings and quadrilaterals are frequently applied in geometric function theory; see, e.g., the handbook by Kühnau [Handbook of Complex Analysis: Geometric Function Theory, Vols. 1 and 2, North–Holland, Amsterdam, 2005]. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new $hp$-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz–Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the $hp$-FEM algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds.

We study convexity and starlikeness of quasihyperbolic and distance ratio metric balls on Banach spaces. In particular, problems related to these metrics on convex domains, and on punctured Banach spaces, are considered

We study properties of quasihyperbolic geodesics on Banach spaces. For example, we show that in a strictly convex Banach space with the Radon-Nikodym property, the quasihyperbolic geodesics are unique. We also give an example of a convex domain $\Omega$ in a Banach space such that there is no geodesic between any given pair of points $x, y \in \Omega\,.$ In addition, we prove that if $\mathrm{X}$ is a uniformly convex Banach space and its modulus of convexity is of a power type, then every geodesic of the quasihyperbolic metric, defined on a proper subdomain of $\mathrm{X}$, is smooth.Comment: 14 pages, 4 figure