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Bounds for an integral of the modified Bessel function of the first kind and expressions involving it
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abstract
Simple upper and lower bounds are obtained for the integral $\int_0^x\mathrm{e}^{-\gamma t}t^\nu I_\nu(t)\,\mathrm{d}t$, $x>0$, $\nu>-\frac{1}{2}$, $0<\gamma<1$. Most of our bounds for this integral are tight as $x\rightarrow\infty$. We apply one of our inequalities to bound some expressions involving this integral. Two of these expressions appear in Stein's method for variance-gamma approximation, and our bounds will allow for a technical advancement to be made to the method.
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$L^p$-Legendre Transforms and Mahler integrals: Asymptotics and the Fokker--Planck heat flow
Introduces the L^p-Legendre transform and proves a functional L^p-Santaló inequality: after translation by the L^p-Santaló point, the L^p-Mahler integral of any convex function is bounded by that of the Gaussian |x|^2/2.
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