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

Abstract

We obtain a simple two-sided inequality for the ratio Lν(x)/Lν−1(x) in terms of the ratio Iν(x)/Iν−1(x), where Lν(x) is the modified Struve function of the first kind and Iν(x) is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for Iν(x)/Iν−1(x) to immediately deduce bounds for Lν(x)/Lν−1(x). We note some consequences and obtain further bounds for Lν(x)/Lν−1(x) by adapting techniques used to bound the ratio Iν(x)/Iν−1(x). We apply these results to obtain new bounds for the condition numbers xLν′(x)/Lν(x), the ratio Lν(x)/Lν(y) and the modified Struve function Lν(x) itself. Amongst other results, we obtain two-sided inequalities for xLν′(x)/Lν(x) and Lν(x)/Lν(y) that are given in terms of xIν′(x)/Iν(x) and Iν(x)/Iν(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.