Let $f,g$ are the functions of $S^{1}$. Are there $f, g$ such that
$$\int\_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int\_{S^{1}}f'g<0,$$
where $f'=\frac{\partial f}{\partial \theta}$?



Are there $f, g$ such that $\int_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int_{S^{1}}f'g<0,$ where $f'=\frac{\partial f}{\partial \theta}$
No, there are no such functions. Indeed, if $h$ is periodic and has mean zero, then $\|h\|\_2 \le \|h'\|\_2$. Take $f,g$ as above and write $g=g\_1+c$ with $g\_1$ having mean zero. Then
$$
\|f'\|\_2^2+\|g'\|\_2^2<2\int\_{S^1}f'g=2 \int\_{S^1} f'g\_1 \le 2\|f'\|\_2\|g\_1'\|\_2=2\|f'\|\_2\|g'\|\_2 \le \|f'\|\_2^2+\|g'\|\_2^2.
$$


