Wirtinger's Inequality

If $y$ has period $2\pi$, $y'$ is $L^2$, and

$$\int_0^{2\pi} y\,dx=0,$$

then

$$\int_0^{2\pi} y^2\,dx<\int_0^{2\pi} y'^2\,dx$$

unless

$$y=A\cos x+B\sin x.$$

References

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 184-187, 1988.