On Robinson’s 1\over2 conjecture

Abstract

In 1947, R. Robinson conjectured that if f is in S, i.e. a normalized univalent function on the unit disk, then the radius of univalence of [ z f ( z ) ] ′ / 2 [zf(z)]’/2 is at least 1 2 \tfrac {1}{2} . He proved in that paper that it was at least .38. The conjecture has been shown to be true for most of the known subclasses of S. This author shows through use of the Grunski inequalities, that the minimum lower bound over the class S lies between .49 and .5.