Higher order Wirtinger inequalities

Abstract

SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

On quadratic integral inequalities of the second order

“…2 Remark 4. The identity (12) was already used by Kreith and Swanson [8]. To derive it the authors considered admissible functions vanishing at the endpoints on the closed interval [a, b].…”

“…Let us notice that the condition ω 0 on U α and v 0 (α) < 0 is possible only if v 2 (α) < 0. Now we shall consider the limit behaviour of the functions v 0 , v 1 and v 2 appearing in types α 2 -α 8 to show under what boundary conditions on function h we obtain lim inf t→α v 0 h 2 0, lim inf t→α v 1 hh 0 and lim inf t→α v 2 h 2 0, respectively.…”

“…For references see, e.g. [2,3,8,10,11]. Further detailed studies of this type of integral inequalities can be found in the monographs [7] and [12].…”