Let $f(t)$ and $g(t)$ be periodic functions on $t\in[0,2\pi]$. By using the Fourier series of the two functions, we can easily prove the inequality
$$\left|\int\_0^{2\pi}f(t)g'(t)dt\right|=
 \left|\int\_0^{2\pi}f'(t)g(t)dt\right|\le
 \frac{1}{2}\int\_0^{2\pi}[f'(t)^2+g'(t)^2]dt\text.$$


I have been trying to find a reference for this inequality because I need to use it to solve some problem. The closest I have been able to find is [Pachpatte 1986](http://www.sciencedirect.com/science/article/pii/0022247X86902283), which gives
$$\frac{1}{2}\int\_0^{2\pi}\left[|f(t)||g'(t)|+|f'(t)||g(t)|\right]dt\le
\frac{\pi}{2}\int\_0^{2\pi}[f'(t)^2+g'(t)^2]dt\text.$$


The extra factor of $\pi$ is highly undesirable and the absolute values inside of
the integral unnecessary for me. I can easily provide a short proof in the text,
but if anybody can think of where the first inequality might appear, that would
be better.



A Wirtinger-like inequality involving two functions
Your inequality is implicit in Hurwitz's Fourier series proof of the isoperimetric inequality in the plane. See for example section 36 of Körner's *Fourier Analysis* or section 4.1 of Groemer's *Geometric Applications of Fourier Series and Spherical Harmonics*.


