The radius of uniform convexity of Bessel functions

In this paper, we determine the radius of uniform convexity for three kinds of normalized Bessel functions of the first kind. In the cases considered the normalized Bessel functions are uniformly convex on the determined disks. Moreover, necessary and sufficient conditions are given for the parameters of the three normalized functions such that they to be uniformly convex in the open unit disk. The basic tool of this study is Bessel functions in series.