Wirtinger's inequality

Statement

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

$\underset{0}{\overset{\pi }{\int }}{(\frac{dF}{dt})}^{2}dt\ge \underset{0}{\overset{\pi }{\int }}F(t{)}^{2}dt$

Applications

• Isoperimetric inequality can be proved using Wirtinger's inequality