A kind of sharp Wirtinger inequalities.

This study gives a kind of sharp Wirtinger inequalities (Pizone inequalities) ∥ f (k) ∥ p ≤ B n , k , p , q (b − a) n − k + 1 / p − 1 / q ∥ f (n) ∥ q for all   1 ≤ p , q ≤ ∞ , 0 ≤ k ≤ n − 1 , where f ∈ 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 , k , p , ∞ to the maximum value problem of a multivariate function, and we give the values of B n , k , p , ∞ by finding the solution of the multivariate function aforementioned. At last, we refer the computation of B n , k , p , q (1 ≤ q < ∞) 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)]. [ABSTRACT FROM AUTHOR]