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, 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.