Integral means of solutions of the one-dimensional Poisson equation with Robin boundary conditions

Consider the one-dimentional Poisson equation $-u^{\mathrm{\prime }\mathrm{\prime }}=f$ on the interval $[-\pi ,\pi ]$, where $f$ is an non-negative integrable function, with Robin boundary conditions $-u^{\mathrm{\prime }}(-\pi )+\alpha u(-\pi )=u^{\mathrm{\prime }}(\pi )+\alpha u(\pi )=0$, where $\alpha >0$ is a constant. In this paper we prove inequalities for the convex integral means of solutions of this problem, using the polarization of functions and its properties. We find solutions with maximal convex integral means and prove their uniqueness.