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Inequality between product of finite integrals

Source: Romanian District Olympiad 2006, Grade 12, Problem 1

March 11, 2006
inequalitiescalculusintegrationfunctionreal analysisreal analysis unsolved

Problem Statement

Let f1,f2,…,fn:[0,1]→(0,∞)f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty) be nn continuous functions, n≥1n\geq 1, and let σ\sigma be a permutation of the set {1,2,…,n}\{1,2,\ldots, n\}. Prove that ∏i=1n∫01fi2(x)fσ(i)(x)dx≥∏i=1n∫01fi(x)dx. \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx.
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