Rounding corners of gearlike domains and the omitted area problem

A function g analytic on the open unit disk D and vanishing only at the origin is said to be gearlike if g maps D to a domain whose boundary consists of arcs of circles centered at the origin and segments of rays emanating from the origin. The authors discuss each of the possible types of (boundary) corners the image domain of gearlike functions may have and give formulae for rounding or smoothing each of these possible corners, extending some early work of P. Henrici. The omitted area problem, first posed by Goodman in 1949, is to determine for a normalized univalent analytic function f on D the maximum area in D which can be omitted from the range of f. While Goodman gave some early bounds for the maximal omitted area, the problem has generally proved to be one of the difficult and long outstanding problems in geometric function theory. The authors apply the method of rounding corners to a specifically constructed gearlike function to produce an approximation for the extremal solution.