A proof of the Khavinson conjecture in \(\mathbb{R}^3\) (Q2313366)

The author proves the Khavinson conjecture in \(\mathbb{R}^3\). In [Can. Math. Bull. 35, No. 2, 218--220 (1992; Zbl 0776.31004)], \textit{D. Khavinson} obtained the sharp pointwise constant in the estimate of the absolute value of the radial derivative of a bounded harmonic function in the unit ball of \(\mathbb{R}^3\). He conjectured the validity of the stronger inequality for the modulus of the gradient of the harmonic function, instead of its radial derivative. In the present paper, the author proves such conjecture showing that \[ |\nabla u(x)| \leq \frac{1}{\rho^2} \Bigg(\frac{(1+\frac{1}{3}\rho^2)^{3/2}}{1-\rho^2}-1\Bigg)\sup_{|y|<1}|u(y)|\, , \] with \(\rho=|x|\) and \(u\) is any bounded harmonic function in the unit ball of \(\mathbb{R}^3\).