Let $u$ be a harmonic function inside the unit ball $B(0, 1)$ in $\mathbb{R}^2$ so that $|u|\leqslant 1$. Does a function u which satisfies $|\nabla u(0)|>1$ exist? If not, please prove that $|\nabla u(0)|\leqslant1$. Thanks.



A question about harmonic function
The answer is "Yes" by the following harmonic analogy of Schwarz lemma:


See Proposition 1.5. of this paper which says that $4/\pi$ is a sharp upper bound.


<https://arxiv.org/pdf/1010.4905.pdf>


