Journal of Inequalities and Applications
Volume 2 (1998), Issue 1, Pages 1-36

Five integral inequalities; an inheritance from Hardy and Littlewood

W. D. Evans,1 W. N. Everitt,2 W. K. Hayman,3 and D. S. Jones4

1School of Mathematics, University of Wales Cardiff, The Mathematical Institute, Senghennydd Road, Wales, Cardiff CF2 4YH, UK
2School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
3Department of Mathematics, Imperial College, Huxley Building, 180 Queen’s Gate, London SW7 2BZ, UK
4Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, Scotland

Received 11 February 1997

Copyright © 1998 W. D. Evans et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 greater detail in the now classic text Inequalities of 1934, written by Hardy and Littlewood together with G. Pólya.

All these inequalities involve Lebesgue square-integrable functions, together with their first two derivatives, integrated over the positive half-line of the real field.

The method to discuss the analytical properties of these inequalities is based on the Sturm–Liouville theory of the underlying second-order differential equation, and the associated Titchmarsh–Weyl m-coefficient.

The five examples are specially chosen so that the corresponding Sturm–Liouville differential equations have solutions in the domain of special functions; in the case of these examples the functions involved are those named as the Airy, Bessel, Gamma and the Weber parabolic cylinder functions. The extensive range of known properties of these functions enables explicit analysis of some of the analytical problems to give definite results in the examples of this paper.

The analytical problems are “hard” in the technical sense and some of them remain unsolved; this position leads to the statement in the paper of a number of conjectures.

In recent years the difficulties involved in the analysis of these problems led to a numerical approach and this method has been remarkably successful. Although such methods, involving standard error analysis and the inevitable introduction of round-off error, cannot by their nature provide analytical proofs; nevertheless the now established record of success of these numerical methods in predicting correct analytical results lends authority to the correctness of the conjectures made in this paper.