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The best constants in the Wirtinger inequality

This study aimed to determine the best constants in the Wirtinger inequality ∥f∥2 ≤ Cr∥f(r)∥ 2, where f is defined on [0, 1] with f(0) = f′(0) = ⋯ = f(r−1)(0) = 0. First, we referred the computation of Cr to the maximal eigenvalue of an integral type operator Ar. Second, we proved that the computation of the eigenvalues of Ar is equivalent to the solution of a Strum–Liouville problem with some boundary conditions and hence we referred the computation of Cr 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 Cr: limr→∞Cr ⋅ r! = 1 2.