A proof of the Khavinson conjecture in R3

This paper deals with an extremal problem for bounded harmonic functions in the unit ball B-n. We solve the Khavinson conjecture in 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: vertical bar del u(x)vertical bar <= 1/rho(2)((1+1/3 rho(2))(3/2)/1-rho 2 -1) sup(vertical bar y vertical bar<1)vertical bar u(y)vertical bar, with rho = vertical bar x vertical bar, thus sharpening the previously known with vertical bar <del u(x), n(x)>vertical bar instead of vertical bar del u(x)vertical bar, where n(x = )x/vertical bar x vertical bar (C) 2019 Elsevier Inc. All rights reserved.