A proof of the Khavinson conjecture

Abstract

\begin{abstract} This paper deals with an extremal problem for bounded harmonic functions in the unit ball ${\mathbb{B}}^n.$ We solve the Khavinson conjecture in ${\mathbb{R}}^3,$ an intriguing open question since 1992 posed by D. Khavinson, later considered in a general context by Kresin and Maz'ya. Precisely, we obtain the following inequality: $$|\nabla u(x)|\le \frac{1}{{\rho }^2}(\frac{(1+\frac{1}{3}{\rho }^2{)}^{\frac{3}{2}}}{1-{\rho }^2}-1)\underset{|y|<1}{\sup }|u(y)|,$$ with $\rho =|x|,$ thus sharpening the previously known with $|⟨\nabla u(x),n_x⟩|$ instead of $|\nabla u(x)|,$ where $n_x=\frac{x}{|x|}.$ \end{abstract}

18 pages; The lemma 6 is corrected and Lemma 7 is added