Inequalities for an integral involving the modified Bessel function of the first kind
Robert E. Gaunt
TL;DR
This work derives sharp bounds for the integral $ \int_0^x e^{- gamma t} t^{\nu} I_{\nu}(t)\,dt $ and its natural generalization, valid for $x>0$, $\nu>-1/2$, and $0\leq\gamma<1$. The central contribution is a main bound with an improved constant, extending previous results from $\nu\ge\tfrac12$ to $\nu\ge 0$ and matching the correct small- and large-$x$ asymptotics; it is complemented by upper and lower bounds for $\int_0^x e^{- gamma t} t^{\nu} I_{\nu+n}(t)\,dt$ and a two-sided framework $L_{\nu,n}(x)$, $U_{\nu,n}(x)$ that are tight in key limits. These results feed into sharper bounds for expressions arising in Stein's method for variance-gamma approximation, improving constants in the VG Stein equation solutions and related bounds. Overall, the paper provides both direct integral inequalities and a broader two-sided structure for Bessel-function-containing integrals with clear implications for probabilistic approximations and analytic inequalities.
Abstract
Simple bounds are obtained for the integral $\int_0^x\mathrm{e}^{-γt}t^νI_ν(t)\,\mathrm{d}t$, $x>0$, $ν>-1/2$, $0\leqγ<1$, together with a natural generalisation of this integral. In particular, we obtain an upper bound that holds for all $x>0$, $ν>-1/2$, $0\leqγ<1$, is of the correct asymptotic order as $x\rightarrow0$ and $x\rightarrow\infty$, and possesses a constant factor that is optimal for $ν\geq0$ and close to optimal for $ν>-1/2$. 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.