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Suppose $f$ and $g$ are non-negative functions such that $\int f(x)dx = \int g(x)dx=1$. Is it true that $$\Big(\int fg \Big)\,\Big(\int f^2 + g^2\Big)\leq \int f^2g+fg^2\quad?$$

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    $\begingroup$ I think the main question would be: why would you think it's true? $\endgroup$
    – Todd Trimble
    Commented Feb 21, 2014 at 2:56

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The answer is NO. Take $f(x) = 2x$ and $g(x) = 2(1-x)$ on $[0,1]$, then $\int fg = 2/3$ and $\int f^2 + g^2 = 8/3$, but $\int f^2g + fg^2 = 4/3$.

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