  The asymptotic behaviour, with respect to the large order, of the radii of
starlikeness of two types of normalised Bessel functions is considered. We
derive complete asymptotic expansions for the radii of starlikeness and provide
recurrence relations for the coefficients of these expansions. The proofs rely
on the notion of Rayleigh sums and asymptotic inversion. The techniques
employed in the paper could be useful to treat similar problems where inversion
of asymptotic expansions is involved.
