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Integral inequality

Let f:[0, 1] \rightarrow \mathbb{R} be a continuous function such that

    \[\int_0^1 f(x) \, \mathrm{d}x = \kappa = \int_0^1 x f(x) \, \mathrm{d}x\]

Prove that \int_0^1 f^2(x) \, \mathrm{d}x \geq 4 \kappa^2.

Solution

We have successively:

    \begin{align*} \int_0^1 f^2(x) \, \mathrm{d}x &= \int_0^1 (3x-1)^2 \, \mathrm{d}x \int_0^1 f(x)^2 \, \mathrm{d}x \\ & \geq \left( \int_0^1 (3x-1)f(x) \, \mathrm{d}x \right)^2 \\ &= (3k-k)^2 \\ &= 4k^2 \end{align*}

Since f is continuous  equality holds only if f(x)=C(3x-1) for some constant C. From the data we get C=2k.

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