Inequalities for an integral involving the modified Bessel function of the first kind
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916575338037248 |
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| author | Gaunt, Robert E. |
| author_facet | Gaunt, Robert E. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12197 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inequalities for an integral involving the modified Bessel function of the first kind Gaunt, Robert E. Classical Analysis and ODEs Primary 33C20, 26D15 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. |
| title | Inequalities for an integral involving the modified Bessel function of the first kind |
| topic | Classical Analysis and ODEs Primary 33C20, 26D15 |
| url | https://arxiv.org/abs/2501.12197 |