Bounds for modified Lommel functions of the first kind and their ratios

Abstract

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.