Integral means of solutions of the one-dimensional Poisson equation with Robin boundary conditions (Q6615999)

Let $\alpha>0$. For $0\leq f \in L^{1}[-\pi,\pi]$, let $u_{f}$ denote the solution to the Robin boundary value problem \N\[\N\begin{cases} \, -u'' = f, \quad\text{on }[-\pi,\pi], \\\N\, -u'(-\pi) + \alpha u(-\pi) = u'(\pi) + \alpha u(\pi) = 0. \end{cases}\N\] \NThe first result presented in the paper states the following. Let $f_{H}$ be the polarization of $f$ (towards zero) with respect to a point $b\in(-\pi,0)\cup(0,\pi)$, and let $u_{f_{H}}$ be the corresponding solution of the Robin problem. Then, for every convex and increasing function $\phi \colon \mathbb{R} \to \mathbb{R}$, the inequality \N\[\N\int_{-\pi}^{\pi} \phi(u_{f}(x))\,\mathrm{d}x \leq \int_{-\pi}^{\pi} \phi(u_{f_{H}}(x))\,\mathrm{d}x \N\]\Nis satisfied.\N\NThe second result is as follows. Let $f^{\#}$ be the symmetric decreasing rearrangement of $f$, and let $u_{f_{H}}$ be the corresponding solution of the Robin problem. Then, for every convex and increasing function $\phi \colon \mathbb{R} \to \mathbb{R}$, the inequality \N\[\N\int_{-\pi}^{\pi} \phi(u_{f}(x))\,\mathrm{d}x \leq \int_{-\pi}^{\pi} \phi(u_{f^{\#}}(x))\,\mathrm{d}x \N\]\Nis satisfied.\N\NFurthermore, the paper provides precise conditions under which the inequalities above are equalities.