Theorem: Let $f\colon\R\to\R$ be a periodic function of period $2\pi$ , which is continuous and has a continuous derivative throughout $\R$ , and such that \begin{equation} \label{eq:1} \int_0^{2\pi}f(x)=0\;. \end{equation}Then \begin{equation} \label{eq:2} \int_0^{2\pi}f'^2(x)dx\ge\int_0^{2\pi}f^2(x)dx \end{equation}with equality if and only if $f(x)=a\cos x+b\sin x$ for some $a$ and $b$ (or equivalently $f(x)=c\sin (x+d)$ for some $c$ and $d$ ).

Proof:Since Dirichlet's conditions are met, we can write $$f(x)=\frac{1}{2}a_0+\sum_{n\ge 1}(a_n\sin nx+b_n\cos nx)$$ and moreover $a_0=0$ by (). By Parseval's identity, $$\int_0^{2\pi}f^2(x)dx=\sum_{n=1}^\infty(a_n^2+b_n^2)$$ and $$\int_0^{2\pi}f'^2(x)dx=\sum_{n=1}^\infty n^2(a_n^2+b_n^2)$$ and since the summands are all $\ge 0$ , we get (), with equality if and only if $a_n=b_n=0$ for all $n\ge 2$ .

Hurwitz used Wirtinger's inequality in his tidy 1904 proof of the isoperimetric inequality.