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Integral inequality

Source: 2023 4th OMpD LU P4 - Brazil - Olimpíada Matemáticos por Diversão

September 21, 2023
calculusinequalitiesintegrationfunction

Problem Statement

Let n≥0n \geq 0 be an integer and f:[0,1]→Rf: [0, 1] \rightarrow \mathbb{R} an integrable function such that: ∫01f(x)dx=∫01xf(x)dx=∫01x2f(x)dx=…=∫01xnf(x)dx=1\int^1_0f(x)dx = \int^1_0xf(x)dx = \int^1_0x^2f(x)dx = \ldots = \int^1_0x^nf(x)dx = 1 Prove that: ∫01f(x)2dx≥(n+1)2\int_0^1f(x)^2dx \geq (n+1)^2
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