In 1932, Hardy and Littlewood  proved the inequality
The constant 4 is best possible; equality occurs when f(x) = A Y(Bx), where
y(x) = e−½x sin (x sin y−y) (y = ⅓π), (x ≧ o)
and A and B (>0) are constants. In , three proofs are given. The inequality has also been discussed in [3, 4]. A very elementary proof in which the function Y(x) emerges naturally is given in this paper.