A Constructive Method for Numerically Computing Conformal Mappings for Gearlike Domains

An algorithm and program are described for computing the conformal mapping function from the unit disk to a gearlike domain, i.e., a domain containing the origin whose boundary consists of arcs of circles centered at the origin and segments of radial lines emanating from the origin. The gearlike domain may be bounded or unbounded and may have slits. Both the forward and the inverse mapping functions are computable. Several examples are illustrated.An application of the program is given which shows that a coefficient conjecture of R. W. Barnard for bounded univalent starlike functions fails.