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

Robert E. Gaunt

Published 2018-03-20Version 1

We obtain a simple two-sided inequality for the ratio $\mathbf{L}_\nu(x)/\mathbf{L}_{\nu-1}(x)$ in terms of the ratio $I_\nu(x)/I_{\nu-1}(x)$, where $\mathbf{L}_\nu(x)$ is the modified Struve function of the first kind and $I_\nu(x)$ is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for $I_\nu(x)/I_{\nu-1}(x)$ to immediately deduce bounds for $\mathbf{L}_\nu(x)/\mathbf{L}_{\nu-1}(x)$. We note some consequences and obtain further bounds for $\mathbf{L}_\nu(x)/\mathbf{L}_{\nu-1}(x)$ by adapting techniques used to bound the ratio $I_\nu(x)/I_{\nu-1}(x)$. We apply these results to obtain new bounds for the condition numbers $x\mathbf{L}_\nu'(x)/\mathbf{L}_\nu(x)$, the ratio $\mathbf{L}_\nu(x)/\mathbf{L}_\nu(y)$ and the modified Struve function $\mathbf{L}_\nu(x)$ itself. Amongst other results, we obtain two-sided inequalities for $x\mathbf{L}_\nu'(x)/\mathbf{L}_\nu(x)$ and $\mathbf{L}_\nu(x)/\mathbf{L}_\nu(y)$ that are given in terms of $xI_\nu'(x)/I_\nu(x)$ and $I_\nu(x)/I_\nu(y)$, respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The results obtained in this paper complement and improve existing bounds in the literature.