Schwarz-Pick Lemma for Invariant Harmonic Functions on the Complex Unit Ball

This paper establishes a sharp Schwarz-Pick type inequality for real-valued invariant harmonic functions defined on the complex unit ball ${\mathbb{B}}^n$. The proof of this main result simultaneously provides a solution to a natural extension of the Khavinson conjecture for invariant harmonic functions, demonstrating that the sharp constants for the gradient and the radial derivative coincide. As further consequences of the main theorem, we derive two corollaries.