5 INTEGRAL-INEQUALITIES - AN INHERITANCE FROM HARDY AND LITTLEWOOD

This paper is concerned with five integral inequalities considered as generalisations of an inequality first discovered by G.H. Hardy and J. E. Littlewood in 1932. Subsequently the inequality was considered in g reater detail in the now classic text Inequalities of 1934, written by Hardy and Littlewood together with G. Polya. All these inequalities i nvolve Lebesgue square-integrable functions, together with their first two derivatives, integrated over the positive half-line of the real f ield. The method to discuss the analytical properties of these inequal ities is based on the Sturm-Liouville theory of the underlying second- order differential equation, and the associated Titchmarsh-Weyl m-coef ficient. The five examples are specially chosen so that the correspond ing Sturm-Liouville differential equations have solutions in the domai n of special functions; in the case of these examples the functions in volved are those named as the Airy, Bessel, Gamma and the Weber parabo lic cylinder functions. The extensive range of known properties of the se functions enables explicit analysis of some of the analytical probl ems to give definite results in the examples of this paper. The analyt ical problems are ''hard'' in the technical sense and some of them rem ain unsolved; this position leads to the statement in the paper of a n umber of conjectures. In recent years the difficulties involved in the analysis of these problems led to a numerical approach and this metho d has been remarkably successful. Although such methods, involving sta ndard error analysis and the inevitable introduction of round-off erro r, cannot by their nature provide analytical proofs; nevertheless the now established record of success of these numerical methods in predic ting correct analytical results lends authority to the correctness of the conjectures made in this paper.