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 alpharightarrowinfty. 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.
Highlights
Ball’s integral inequality [ ], in connection with cube-slicing in Rn, says that for all s ≥, ∞ sin(π x) s dx ≤, or–∞ π x s sin x s dx ≤ π–∞ x s with strict inequality except when s =
It suggests that the integral decays like
We showed that the asymptotic behavior I(α, [α]– )
Summary
Ball’s integral inequality [ ], in connection with cube-slicing in Rn, says that for all s ≥ ,. –∞ x s with strict inequality except when s =. It suggests that the integral decays like. < , the asymptotic result implies the inequality for large values of s. The asymptotic result, though reasonably tame, presents new difficulties when we consider a more general integral, and this is circumvented here by the proof of two new inequalities. The inequalities obtained are indispensable in obtaining the asymptotic behavior, especially in the interesting “oscillatory” cases. The oscillatory behavior makes it impossible to employ the standard methods used in connection with Ball’s inequality.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
