A kind of sharp Wirtinger inequalities

Abstract

This study gives a kind of sharp Wirtinger inequalities (Pizone inequalities) [Formula: see text] where [Formula: see text] with at least [Formula: see text] zeros (counting multiplicity) in [Formula: see text]. First, based on the Hermite (Lagrange) interpolation, we express [Formula: see text] as a Lagrange type (integral type) remainder. Second, we refer the computation of [Formula: see text] to the maximum value problem of a multivariate function, and we give the values of [Formula: see text] by finding the solution of the multivariate function aforementioned. At last, we refer the computation of [Formula: see text] to the norm of an integral operator. Our results are corrections and extensions to the results that appear in [J. C. Kuang, Applied Inequalities (Shandong Science and Technology Press, Jinan, 2004); A. Yu. Levin, Some estimates for a diff erentiable function, Dokl. Akad. Nauk SSSR 138 (1961) 37-38 (in Russian)].