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Periodic integral

Source: IberoAmerican Olympiad For University Students

July 10, 2011
calculusintegrationinequalitiesfunctionlogarithmsreal analysisIntegral inequality

Problem Statement

Let f:R→R+f:\mathbb{R}\to\mathbb{R}^+ be a continuous and periodic function. Prove that for all α∈R\alpha\in\mathbb{R} the following inequality holds:
∫0Tf(x)f(x+α)dx≥T\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T,
where TT is the period of f(x)f(x).
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