A proof of the Khavinson conjecture (Q2035589)

The author gives a complete proof of the validity of the Khavinson conjecture. In order to state the conjecture, let \(h^\infty\) be the space of bounded harmonic functions on the unit ball \(\mathbb{B}^n\) of \(\mathbb{R}^n\), with \(n \geq 3\). For \(x \in \mathbb{B}^n\) we denote by \(C(x)\) the smallest number such that \[ |\nabla u(x)| \leq C(x)\sup_{y \in \mathbb{B}^n}|u(y)| \] for all \(u \in h^\infty\). Similarly, for \(x\in \mathbb{B}^n\) and \(l\in \partial \mathbb{B}^n\), we denote by \(C(x,l)\) the smallest number such that \[ |\langle\nabla u(x),l \rangle | \leq C(x,l)\sup_{y \in \mathbb{B}^n}|u(y)| \] for all \(u \in h^\infty\). As it is well known, both constants are finite. The Khavinson conjecture states that for \(x \in \mathbb{B}^n \setminus \{0\}\) we have \[ C(x)=C\left(x,\frac{x}{|x|}\right)\, . \] The author shows the validity of the conjecture, by considering an equivalent optimization problem and by solving such a problem in terms of the Gegenbauer polynomials.