The radius of univalence of Bessel functions I

It is well known that J,(z) is an entire function for any u. With respect to the normalization factor z1-", we note that it is unique in the sense that 1 u is the only exponent for which z-"J,(z) is schlicht in some neighborhood of the origin when u > -1. The index , is assumed to be real. We present here a complete solution for u > -1. In 5 we state some results for u < --1 which appear plausible in the light of our computational experiments; we expect to handle these in a later paper.