Bounds for an Integral Involving the Modified Lommel Function of the First Kind

Abstract

AbstractSimple upper and lower bounds are established for the integral $$\int _0^x\mathrm {e}^{-\beta u}u^\nu t_{\mu ,\nu }(u)\,\mathrm {d}u$$ ∫ 0 x e - β u u ν t μ , ν ( u ) d u , where $$x>0$$ x > 0 , $$0<\beta <1$$ 0 < β < 1 , $$\mu +\nu >-2$$ μ + ν > - 2 , $$\mu -\nu \ge -3$$ μ - ν ≥ - 3 , and $$t_{\mu ,\nu }(x)$$ t μ , ν ( 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.