An antimaximum principle for periodic solutions of a forced oscillator

Consider the equation of the linear oscillator $u\mathrm{"}+u=h(\theta )$, where the forcing term $h:\mathbb{R}\to \mathbb{R}$ is $2\pi$-periodic and positive. We show that the existence of a periodic solution implies the existence of a positive solution. To this aim we establish connections between this problem and some separation questions of convex analysis.