The purpose of this note is to provide integral inequalities which are related to Hardy's ( and [3, Theorem 330]). This latter result we state as
Theorem 1. Let p>1, r≠1, and ƒ(x) be nonnegative and Lebesgue integrable on [0, a] or [a, ∞] for every a>0, according as r> 1 or r< 1. If F(x) is defined by
unless f≡0. The constant is the best possible.