The geometry of convex curves tending to $1$ in the unit disc

Abstract:

Let $f$ be a real convex function defined near ${0^ + },f(0) = f’(0) = 0,f > 0$ otherwise. The family of curves, $f(\theta ) = \gamma (1 - r),0 < \gamma < \infty$, in the unit disc is investigated. The union of the ${w^ \ast }$ closures in the maximal ideal space of ${H^\infty }$ of these curves is seen to be a union of nontrivial parts and each such part is hit by the closure of each such curve. Using results of Hoffman and the Wermer map onto such a part, nets in the disc tending to points of the part are located on the curves as explicitly as can be expected. One striking consequence of this coordinatization is the fact that the closure of any such curve in a part is invariably the image under the Wermer map of an oricycle. This is not true for the limiting case of the family of Stolz rays at 1. Other detailed results of a similar nature are derived.