Inequalities for Some Integrals Involving Modified Lommel Functions of the First Kind

Abstract

In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind t µ,ν (x). In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality, involving the modified Lommel function t µ,ν (x), for a generalized hypergeometric function. The inequalities obtained in this paper generalise recent bounds for integrals involving the modified Struve function of the first kind.

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

“…When specialised to the case of the modified Struve function L ν (x), our bounds are sharper than those of [17] (see Corollary 2.6). We also establish several lower bounds for the integral (1.2) (Theorem 2.4), one of which is strictly sharper than the only lower bound given in [13]. In fact, all lower bounds derived in this paper are tight in the limit x → ∞, a property not enjoyed by the lower bound of [13].…”

“…We also establish several lower bounds for the integral (1.2) (Theorem 2.4), one of which is strictly sharper than the only lower bound given in [13]. In fact, all lower bounds derived in this paper are tight in the limit x → ∞, a property not enjoyed by the lower bound of [13]. When specialised to the case of the modified Struve function L ν (x), one of the lower bounds is sharper than that of [17] (see Corollary 2.6).…”

“…The condition µ > − 3 2 , − 1 2 < ν < µ + 1 ensures that µ + ν > −2 (so that the integral exists), ν > − 1 2 and µ − ν > −1. When 0 ≤ β < 1, an exact formula for the integral (1.2) in terms of the functiont µ,ν (x) is not available, although the integral can be evaluated exactly in terms of the generalized hypergeometric function (the case β = 0) or an infinite series involving lower incomplete gamma functions (when 0 < β < 1); see [13]. The fact that no simple closed-form formulas are available for the integral (1.2) provides the motivation for establishing simple bounds, involving the functiont µ,ν (x) itself, for this integral.…”

“…Several upper bounds and a lower bound for the integral (1.2) were established by [13]. 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).…”

“…In Theorem 2.2, we extend the range of validity (from µ > − 1 2 , 1 2 ≤ ν < µ + 1 to µ > − 3 2 , − 1 2 < ν < µ+1) of the upper bounds of [13] for the integral (1.2), with our bounds having the same functional form, but larger numerical constants. When specialised to the case of the modified Struve function L ν (x), our bounds are sharper than those of [17] (see Corollary 2.6).…”