The Modulus of the Image Annuli Under Univalent Harmonic Mappings and a Conjecture of Nitsche

The object of the paper is to show that if f is a univalent,harmonic mapping of the annulus A(r, 1) = {z : r < |z| <1} onto the annulus A(R, 1), and if s is the length of the segmentof the Grötzsch ring domain associated with A(r, 1), thenR < s. This gives the first, quantitative upper bound ofR, which relates to a question of J. C. C. Nitsche that he raisedin 1962. The question of whether this bound is sharp remainsopen.