  The modified Lommel function $t_{\mu,\nu}(x)$ is an important special
function, but to date there has been little progress on the problem of
obtaining functional inequalities for $t_{\mu,\nu}(x)$. In this paper, we
advance the literature substantially by obtaining a simple two-sided inequality
for the ratio $t_{\mu,\nu}(x)/t_{\mu-1,\nu-1}(x)$ in terms of the ratio
$I_\nu(x)/I_{\nu-1}(x)$ of modified Bessel functions of the first kind, thereby
allowing one to exploit the extensive literature on bounds for this ratio. We
apply this result to obtain two-sided inequalities for the condition numbers
$xt_{\mu,\nu}'(x)/t_{\mu,\nu}(x)$, the ratio $t_{\mu,\nu}(x)/t_{\mu,\nu}(y)$
and the modified Lommel function $t_{\mu,\nu}(x)$ itself that are given in
terms of $xI_\nu'(x)/I_\nu(x)$, $I_\nu(x)/I_\nu(y)$ and $I_\nu(x)$,
respectively. The bounds obtained in this paper are quite accurate and often
tight in certain limits. As an important special case we deduce bounds for
modified Struve functions of the first kind and their ratios, some of which are
new, whilst others extend the range of validity of some results given in the
recent literature.
