On an inequality of the Kolmogorov type for a second-order differential expression

Abstract

In this paper we discuss an integro-differential inequality formed from the square of a second-order differential expression. A connection between the existence of the inequality and the Titchmarsh–Weyl m -function is established and it is shown that the best constant in the inequality is determined by the behaviour of the m -function. Analytic results are established for specific classes of problems. A numerical treatment of the inequality is also discussed and estimates of the best constant in specific examples are given.