On two integral inequalities

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Abstract
In 1932, Hardy and Littlewood [1] proved the inequalityThe constant 4 is best possible; equality occurs when f(x) = A Y(Bx), wherey(x) = e−½x sin (x sin y−y) (y = ⅓π), (x ≧ o)and A and B (>0) are constants. In [2], 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.

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