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A general version of the Hardy–Littlewood–Polya–Everitt (HELP) inequality

Synopsis The inequality (0·1) below is naturally associated with the equation −(pu′)′ + qu = λu. By assuming that one end-point of the interval (a, b) is regular and the other limit-point for this equation, Everitt characterized the best constant K in tems of spectral properties of the equation. This paper sketches a theory for more general inequalities (0·2), (0·3) similarly related to the equation Su = λTu. Here S and T are ordinary, symmetric differential expressions. A characterization of the best constants in (0·2), (0·3) is given which generalises that of Everitt. For the case when S is of order 1 and T is multiplication by a positive function, all possible inequalities are given together with the best constants and cases of equality. Furthermore, an example is given of a valid inequality (0·1) on an interval with both end-points regular for the corresponding differential equation. This contradicts a conjecture by Everitt and Evans. Finally, the general theory for the left-definite inequality (0·3) is specialised to the case when S is a Sturm-Liouville expression. A family of examples is given for which the best constants can be explicitly calculated.