On inequalities of Hilbert and Widder

These inequalities are equivalent, and they are known as Widder's inequalities. Notethat (1) is stronger than the well-known Hilbert inequality (see [4]). G. H. Hardy [2]showed that (2) (and so (1)) can be obtained by using the same Hilbert inequality (seealso [3, pp. 238-239]).The following generalization of Widder's inequality is given by Y. C. Chow [1]:Theorem 1. Let a and b be nonnegative two sequences, p>l, p' = p/{p—l). Then