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for questions involving inequalities, upper and lower bounds.

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Is this Holder-type inequality true $\;(\int fg )\,(\int f^2 + g^2)\leq \int f^2g+fg^2\;$? [closed]

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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