Wirtinger's inequality

Statement

Let $F:[0,\pi ]\to \mathbb {R}$ be a smooth function such that $F(0)=F(\pi )=0$. Then:

$\int _{0}^{\pi }\left({\frac {dF}{dt}}\right)^{2}dt\geq \int _{0}^{\pi }F(t)^{2}dt$

Applications

• Isoperimetric inequality can be proved using Wirtinger's inequality