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Consider $f,g$ where $f(x) > g(x) \ge 0 \ \forall x \in (0,1)$ and $f(0) = g(0)$, $f(1) = g(1)$ . Is the following inequality true?

$\int_0^t \left[f(x)-g(x) + l\right]\mathrm dx > \int_t^1 \left[f(x-t)-f(x)\right] \mathrm dx$

for any $l > 0, 0

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The inequality is not true. For a counter-example (for sure not the easiest) consider $f(x) = 1 -\frac{x}{2},$ $g(x)=f(x) - x(1-x)$ and $l=\frac{1}{24}$ and $t=\frac{1}{2}$. Then the left hand side is $\frac{5}{48}$ and the right hand side is $\frac{1}{8} = \frac{6}{48}$.

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    @measure_noob: D'oh. I completely misread that, over and over and over again! :-D. Fabian: Terribly sorry :-)2011-05-07