  In this paper our aim is to deduce some sufficient (and necessary) conditions
for the close-to-convexity of some special functions and their derivatives,
like Bessel functions, Struve functions, and a particular case of Lommel
functions of the first kind, which can be expressed in terms of the
hypergeometric function ${}_1F_2$. The key tool in our proofs is a result of
Shah and Trimble about transcendental entire functions with univalent
derivatives. Moreover, a known result of P\'olya on entire functions, the
infinite product representations and some results on zeros of Bessel, Struve
and Lommel functions of the first kind are used in order to achieve the main
results of the paper.
