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

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

The answer is "Yes" by the following harmonic analogy of Schwarz lemma:

See Proposition 1.5 of this paper

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

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