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There is a question that, I think, has a definite answer, but I can't figure it out. Given are three real valued functions, $f$,$g$, $w$, of a real variable $x$. The functions are non-negative, i.e., $f(x) \ge 0$, $g(x) \ge 0$ and $w(x) \ge 0$ for all $x$. We have also that $\sqrt{w} \le \sqrt{f} + \sqrt{g}$ for all $x$ and that

\begin{equation} \intop_{a}^{b}w(x)dx = \intop_{a}^{b}f(x)dx + \intop_{a}^{b}g(x)dx \end{equation}

where $a$ and $b$ are finite. The functions also obey periodic boundary conditions, i.e., $w(a) = w(b)$, $f(a) = f(b)$ and $g(a) = g(b)$.

Can one prove that the expression

\begin{equation} \intop_{a}^{b}(w^2 - f^2 - g^2)dx \end{equation}

is always either non-positive or non-negative?

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