A proof of the Khavinson conjecture

\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: $|\nabla u(x)|\leq \frac{1}{ρ^2}\bigg({\frac{(1+\frac{1}{3}ρ^2)^{\frac{3}{2}}}{1-ρ^2}-1}\bigg) \sup_{|y|<1} |u(y)|,$ with $ρ=|x|,$ thus sharpening the previously known with $|\langle \nabla u(x),n_{x} \rangle |$ instead of $|\nabla u(x)|,$ where $n_{x}=\frac{x}{|x|}.$ \end{abstract}

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