I would like to maximize
$$
\int_0^{2\pi} \frac{(f'(x))^2}{f(x)}dx
$$
subject to $f(x)\leq 1$ for all $x$
over the space of nonnegative trigonometric polynomials of degree smaller or equal to $n$.

The application would be to design a (theoretically) optimal 1d-image for position tracking via template matching.