Inequalities and asymptotics for some moment integrals

• Main Author: Faruk Abi-Khuzam
• Format: Article
• Language: English
• Published: SpringerOpen 2017-10-01
• Series: Journal of Inequalities and Applications
• Subjects: Ball’s inequality; asymptotics; cube slicing; moments
• Online Access: http://link.springer.com/article/10.1186/s13660-017-1532-7

Abstract For α > β − 1 > 0 $\alpha>\beta-1>0$ , we obtain two-sided inequalities for the moment integral I ( α , β ) = ∫ R | x | − β | sin x | α d x $I(\alpha,\beta)=\int_{\mathbb{R}}|x|^{-\beta}|\sin x|^{\alpha}\,dx$ . These are then used to give the exact asymptotic behavior of the integral as α → ∞ $\alpha\rightarrow\infty$ . The case I ( α , α ) $I(\alpha,\alpha)$ corresponds to the asymptotics of Ball’s inequality, and I ( α , [ α ] − 1 ) $I(\alpha,[\alpha]-1)$ corresponds to a kind of novel “oscillatory” behavior.