Bounds for an integral involving the modified Lommel function of the first kind

TL;DR

This paper derives improved upper and lower bounds for an integral involving the modified Lommel function of the first kind, extending and sharpening previous bounds and unifying results with those for the modified Struve function.

Contribution

It provides new, sharper bounds for a specific integral involving the modified Lommel function, broadening the range of validity and connecting to bounds for the modified Struve function.

Findings

Bounds are sharper than existing ones.

Range of validity for bounds is extended.

Connections to bounds for the modified Struve function.

Abstract

Simple upper and lower bounds are established for the integral $\int_0^x\mathrm{e}^{-\beta u}u^\nu t_{\mu,\nu}(u)\,\mathrm{d}u$, where $x>0$, $0<\beta<1$, $\mu+\nu>-2$, $\mu-\nu\geq-3$, and $t_{\mu,\nu}(x)$ is the modified Lommel function of the first kind. Our bounds complement and improve on existing bounds for this integral, by either being sharper or increasing the range of validity. Our bounds also generalise recent bounds for an integral involving the modified Struve function of the first kind, and in some cases more a direct approach lead to sharper bounds when our general bounds are specialised to the modified Struve case.