I find a interesting inequality. Suppose that $y=y(x)$ is a differentiable function in $(0,L)$ and $y(0)=y(a)=0$. Consider the fraction $$ F[y]=\frac{\int_0^{L}\vert y'\vert^2dx}{\int_0^L\vert y\vert^2 dx} $$ where $y'=dy/dx$ and the denominator is nonzero since $y$ is nonezero function. So what's the low bound of $F[y]$? The best constant may be $\frac{\pi^2}{L^2}$, but how to prove it?
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1$\begingroup$ This looks like a job for Calculus of Variations. $\endgroup$– Empy2Commented Nov 27, 2014 at 15:22
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2$\begingroup$ It looks like the celebrated Poincaré-Wirtinger inequality... $\endgroup$– SiminoreCommented Nov 27, 2014 at 15:26
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$\begingroup$ @Michael: Yes, good idea. $\endgroup$– Roger209Commented Nov 27, 2014 at 15:33
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$\begingroup$ @Siminore: Yes, thank you for your hint. $\endgroup$– Roger209Commented Nov 27, 2014 at 15:33
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