Inequalities and asymptotics for some moment integrals

摘要

For α  β − 1  0, we obtain two-sided inequalities for the moment integral I(α, β) = ∫ℝ|x|^−β|sinx|^α dx. These are then used to give the exact asymptotic behavior of the integral as α → ∞. The case I(α, α) corresponds to the asymptotics of Ball’s inequality, and I(α, [α]−1) corresponds to a kind of novel “oscillatory” behavior.