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Holder like inequality

Source: 2019 South Korea USMC P6

August 13, 2020
real analysisinequalitiesIntegral inequalitycollege contestsfunction

Problem Statement

A function f:[0,∞)→[0,∞)f:[0,\infty)\to[0,\infty) is integrable and \int_0^\infty f(x)^2 dx<\infty,  \int_0^\infty xf(x) dx <\infty Prove the following inequality. (∫0∞f(x)dx)3≤8(∫0∞f(x)2dx)(∫0∞xf(x)dx)\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)
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