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

Abstract

We obtain a simple two-sided inequality for the ratio L ν

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

“…(Tables for the case µ = ν are given in [21].) In agreement with the above analysis, we see that, as a result of the exponential decay of b µ,ν (x), the bounds are very accurate for larger values of x, particularly for smaller values of µ, whilst both bounds improve for 'small' x as ν increases.…”

“…We therefore progress the literature from having no bounds for the ratio t µ,ν (x)/t µ−1,ν−1 (x) to a wide variety, and we note some examples. As a special case of Theorem 3.4, we obtain the same two-sided inequality for the ratio L ν (x)/L ν−1 (x) that was obtained in Theorem 2.1 of [21], but with a larger range of validity for the lower bound. This arises from an alternative method of proof that avoids the use of integral representations of the functions L ν (x) and I ν (x) which were used to prove Theorem 2.1 of [21].…”

“…In this paper, we complement that work by obtaining new bounds for the integral (1.2) that are either sharper or have a larger range of validity than those of [13]. We achieve our bounds by adapting the approach used in the recent paper [17] in which similar improvements were obtained on the bounds of [11] for the integral (1.3) involving L ν (x). This approach is fruitful because the properties of the modified Struve function L ν (x) used by [17] generalise naturally to the modified Lommel functiont µ,ν (x).…”

“…, respectively. For space reasons, we omit these corollaries, and just record that the range of validity of inequalities (3.34) and (3.43) of [21] are extended in this paper. On the other hand, for µ = ν the range of validity of inequalities (3.38), (4.44), (4.52) and (4.56) are the same as those for the corresponding inequalities of [21].…”

“…Up to a multiplicative constant, the modified Lommel function t µ,ν (x) generalises the modified Struve function of the first kind L ν (x) (see (1.5)). Therefore, the problem of asking for simple bounds for the integral (1.2) is also a natural generalisation of the problem of obtaining simple bounds, in terms of the modified Struve function of the first kind, for the integral x 0 e −βu u ν L ν (u) du, (1.3) where x > 0, 0 ≤ β < 1, ν > −1, that was recently studied by [11,17].…”