Behaviour of $L_{q}$ norms of the $\sinc_{p}$ function

TL;DR

This paper investigates the asymptotic behavior of $L_q$ norms of the generalized $ ext{sinc}_p$ function, extending classical inequalities and providing new insights into their dependence on $q$ as it approaches infinity.

Contribution

It introduces asymptotic analysis of $L_q$ norms for $ ext{sinc}_p$ functions using recent inequalities involving $p$-trigonometric functions, extending Ball's integral inequality.

Findings

Asymptotic behavior of $L_q$ norms as $q o \infty$ is characterized.

Generalization of Ball's integral inequality to $p$-trigonometric functions.

New inequalities involving $p$-trigonometric functions are utilized.

Abstract

An integral inequality due to Ball involves the $L_{q}$ norm of the $\sinc_p$ function; the dependence of this norm on $q$ as $q\rightarrow\infty$ is now understood. By use of recent inequalities involving $p-$trigonometric functions $(1<p<\infty )$ we obtain asymptotic information about the analogue of Ball's integral when $\sin$ is replaced by $\sin_{p}.$