Abstract:Simple upper and lower bounds are established for the integral $\int _0^x\mathrm {e}^{-\beta t}t^\nu \mathbf {L}_\nu (t)\,\mathrm {d}t$, where $x>0$, $\nu >-1$, $0<\beta <1$ and $\mathbf {L}_\nu (x)$ is the modified Struve function of the first kind. These bounds complement and improve on existing results, through either sharper bounds or increased ranges of validity. In deriving our bounds, we obtain some monotonicity results and inequalities for products of the modified Struve function of the first kind and the modified Bessel function of the second kind $K_{\nu }(x)$, as well as a new bound for the ratio $\mathbf {L}_{\nu }(x)/\mathbf {L}_{\nu -1}(x)$.

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中文翻译：

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涉及第一类修正 Struve 函数的积分的边界

摘要：为积分$\int _0^x\mathrm {e}^{-\beta t}t^\nu \mathbf {L}_\nu (t)\,\mathrm { 建立简单的上下界d}t$, 其中 $x>0$, $\nu >-1$, $0<\beta <1$ 和 $\mathbf {L}_\nu (x)$ 是第一类修正的 Struve 函数. 这些界限通过更清晰的界限或增加的有效性范围来补充和改进现有结果。在推导我们的边界时，我们获得了第一类修正 Struve 函数和第二类修正贝塞尔函数 $K_{\nu }(x)$ 的乘积的一些单调性结果和不等式，以及比率 $\mathbf {L}_{\nu }(x)/\mathbf {L}_{\nu -1}(x)$。

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