On the coefficients of meromorphic schlicht functions

Abstract

looked very convincing at the time. It has, however, recently been shown that this conjecture is false, at least for odd n [1; 3]. Garabedian and Schiffer [1] succeeded, moreover, in proving that the exact value of Aa is 1/2+e-6. While this disposes of the conjecture (n+1)An= 2 in the case of the general class S, one may nevertheless attempt to save the inequality (n+1) I an| g 2 by imposing suitable restrictions on the class S. Taking our cue from a somewhat similar situation which arose in the early discussion of the Bieberbach conjecture, we are led to the consideration of two particular sub-classes of S: (a) the class of functions f(z) with real coefficients an; (b) the class St of functions f(z) which map I z I < 1 onto the complement of a point-set starlike with respect to the origin. The case (a), however, is ruled out immediately, since the Garabedian-Schiffer extremal function happens to have real coefficients. We are thus left with the case (b). It may be remarked that the choice of the origin as the star center of the map appears natural in view of the fact that, because of (1),