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arXiv · 2501.12197

Inequalities for an integral involving the modified Bessel function of the first kind

Robert E. Gaunt
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Abstract

Simple bounds are obtained for the integral $\int_0^x\mathrm{e}^{-\gamma t}t^\nu I_\nu(t)\,\mathrm{d}t$, $x>0$, $\nu>-1/2$, $0\leq\gamma<1$, together with a natural generalisation of this integral. In particular, we obtain an upper bound that holds for all $x>0$, $\nu>-1/2$, $0\leq\gamma<1$, is of the correct asymptotic order as $x\rightarrow0$ and $x\rightarrow\infty$, and possesses a constant factor that is optimal for $\nu\geq0$ and close to optimal for $\nu>-1/2$. We complement this upper bound with several other upper and lower bounds that are tight as $x\rightarrow0$ or as $x\rightarrow\infty$, and apply our results to derive sharper bounds for some expressions that appear in Stein's method for variance-gamma approximation.

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Robert E. Gaunt. 2025-01-21. Inequalities for an integral involving the modified Bessel function of the first kind. https://arxiv.org/abs/2501.12197

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