Some Extensions of a Theorem of Hardy, Littlewood and Pólya and Their Applications

Abstract

In [6], by means of convex functions Φ :R→R, Hardy, Littlewood and Pólya proved a theorem characterizing the strong spectral order relation for any two measurable functions which are defined on a finite interval and which they implicitly assumed to be essentially bounded (cf. [6, the approximation lemma on p. 150 and Theorem 9 on p. 151 of their paper]; see also L. Mirsky [10, pp. 328-329] and H. D. Brunk [1,Theorem A, p. 820]).