Schwarz–Pick inequality for bounded harmonic functions on the unit ball

Let $n=2$ or $n\geq 4$, let $u$ satisfy the hypotheses of Theorem (i), and let $x\in\mathbb{B}_n$. Schwarz–Pick conjecture. One has

$$|\nabla u(x)| \leq \frac {2m_{n-1}(\mathbb{B}_{n-1})}{m_n(\mathbb{B}_n)} \frac{1-|u(x)|^2}{1-|x|^2}.$$

This proposed refinement combines the sharp gradient estimate for bounded harmonic functions on the ball with the classical Schwarz–Pick-type estimate in dimension two. Its status is not resolved in the supplied source.