A kind of sharp Wirtinger inequalities

where f E W-q(n) [a, b] with at least n zeros (counting multiplicity) in [a, b]. First, based on the Hermite (Lagrange) interpolation, we express f as a Lagrange type (integral type) remainder. Second, we refer the computation of B-n,B-k,B-p,B-8 to the maximum value problem of a multivariate function, and we give the values of B(n,k,p,8)by finding the solution of the multivariate function aforementioned. At last, we refer the computation of Bn,k,p,q(1 = q < 8) 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)].