The best constant in an integral inequality

Question

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?

Answer 1

The link is useful:Wirtinger's inequality for functions