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

Abstract

In this paper we discuss an inequality of Kolmogorov type for the square of a second-order formally symmetric difference expression in the limit point case. A connection between the existence of the inequality and the Hellinger–Nevanlinna function associated with the difference expression is established and it is shown that the best constant in the inequality is determined by the behaviour of the -function. Analytical and computational results are obtained for specific classes of problems. Also necessary and sufficient conditions for the powers of the difference expression to be partially separated are given.

Keywords

• Formally symmetric difference expressions
• Kolmogorov type inequalities
• Hellinger&#8211;Nevanlinna <it>m</it>-function
• Limit-point condition
• Partial separation