A proof of the Khavinson conjecture in [formula omitted

Abstract

This paper deals with an extremal problem for bounded harmonic functions in the unit ball Bn. We solve the Khavinson conjecture in R3, 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:|∇u(x)|≤1ρ2((1+13ρ2)321−ρ2−1)sup|y|<1⁡|u(y)|, with ρ=|x|, thus sharpening the previously known with |〈∇u(x),nx〉| instead of |∇u(x)|, where nx=x|x|.