Best constants in a class of integral inequalities

David W. Boyd

Vol. 30 (1969), No. 2, 367–383

DOI: 10.2140/pjm.1969.30.367

Abstract

In this paper a method is developed for determining best constants in inequalities of the following form:

where y(a) = y′(a) = ⋯ = y⁽ⁿ⁻¹⁾(a) = 0 and y⁽ⁿ⁻¹⁾ is absolutely continuous.

It is first shown that for a certain class of m and w, equality can be attained in the inequality. Applying variational techniques reduces the determination of the best constant to a nonlinear eigenvalue problem for an integral operator. If m and w are sufficiently smooth this reduces further to a boundary value problem for a differential equation. The method is illustrated by determinin g the best constants in case (a,b) is a finite interval, m(x) ≡ w(x) ≡ 1, and n = 1.

Mathematical Subject Classification

Primary: 26.70

Milestones

Received: 27 September 1968

Published: 1 August 1969