Let $f$ be $C^1$ in $[-\pi, \pi]$ and satisfies $\int_{-\pi}^\pi f(x)dx=0$, periodic boundary condition.
Then, prove that $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$.
I try to prove it by Parseval's equality(with $X_n=\sin nx$) and Schwartz inequality,
but then some constants come out. Also why condition $\int_{-\pi}^\pi f(x)dx=0$ needs?
Give some advice.
On
Consider the function ${f(x)-f'(x)}^2$ . Integrate this function from $-\pi$ to $\pi$ . As this is a positive function this definite integral is greater than or equal to zero . Now expand the ${f(x)-f'(x)}^2$ using $(a-b)^2$ . In the question he meant that $f(x)$ is an odd function which means $f'(x)$ is an even function . The product $f(x).f'(x)$ is also an odd function and when you integrate, this the term becomes zero . Arrange the other two terms you will get the result . :)
This is known as Wirtinger's inequality. See the wikipedia article for its proof.