I need help in proving this question:
Suppose the function $f$ is differentiable and $\frac{1}{2}$-periodic (i.e., more than just 1-periodic). Show that the Poincare inequality holds with a better constant: $$\int_0^1 f^2(x)dx-\left(\int_0^1 f(x)dx\right)^2\leq \pi^2\int (f'(x))^2dx.$$