This paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) and
The extension consists in replacing f′ by M[f] where
choosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the form
where K is a positive number independent of f.
The analysis involves a fourth-order differential equation and the second-order equation associated with M.
A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.