# Inequalities and Asymptotics for some Moment Integrals

## TL;DR

This paper derives inequalities and asymptotic behaviors for a specific class of moment integrals involving sine functions and power weights, revealing new oscillatory asymptotics as parameters grow large.

## Contribution

It provides two-sided inequalities and exact asymptotics for the moment integral I(α,β), including special cases related to Ball's inequality and oscillatory behavior.

## Key findings

- Established bounds for I(α,β) for α > β-1 > 0
- Derived asymptotic formulas as α approaches infinity
- Connected results to known inequalities and novel oscillatory phenomena

## 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.