  In this paper we determine the radius of convexity for three kind of
normalized Bessel functions of the first kind. In the mentioned cases the
normalized Bessel functions are starlike-univalent and convex-univalent,
respectively, on the determined disks. The key tools in the proofs of the main
results are some new Mittag-Leffler expansions for quotients of Bessel
functions of the first kind, special properties of the zeros of Bessel
functions of the first kind and their derivative, and the fact that the
smallest positive zeros of some Dini functions are less than the first positive
zero of the Bessel function of the first kind. Moreover, we find the optimal
parameters for which these normalized Bessel functions are convex in the open
unit disk. In addition, we disprove a conjecture of Baricz and Ponnusamy
concerning the convexity of the Bessel function of the first kind.
