The best constants in the Wirtinger inequality

Abstract

This study aimed to determine the best constants in the Wirtinger inequality [Formula: see text] where [Formula: see text] is defined on [Formula: see text] with [Formula: see text]. First, we referred the computation of [Formula: see text] to the maximal eigenvalue of an integral type operator [Formula: see text]. Second, we proved that the computation of the eigenvalues of [Formula: see text] is equivalent to the solution of a Strum–Liouville problem with some boundary conditions and hence we referred the computation of [Formula: see text] to finding the minimal zero of a function with one variable. Third, by comparing with a result of Lifshits, Papageorgiou, Woźniakowski, we obtained that the strong asymptotic order of [Formula: see text]: [Formula: see text].