Inequalities and Asymptotics for some Moment Integrals

Abstract

For $\alpha >\beta -1>0$, we obtain two sided inequalities for the moment integral $I(\alpha ,\beta )={\int }_{\mathbb{R}}|x{|}^{-\beta }|\sin x{|}^{\alpha }dx$. These are then used to give the exact asymptotic behavior of the integral as $\alpha \to \infty$. The case $I(\alpha ,\alpha )$ corresponds to the asymptotics of Ball's inequality, and $I(\alpha ,[\alpha ]-1)$ corresponds to a kind of novel "oscillatory" behavior.