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

TL;DR

This work establishes a sharp Schwarz-Pick type inequality for real-valued invariant harmonic functions on the complex unit ball $\mathbb B^n$, showing that $\|\nabla h(z)\| \le \dfrac{2\,\Gamma(n+1)}{\sqrt{\pi}\,\Gamma(n+\tfrac{1}{2})}\dfrac{1}{1-\|z\|^2}$ for all $z$ with $|h|\le 1$, and that the bound is attained in the radial direction, thereby solving an invariant-harmonic version of the Khavinson conjecture. The proof combines explicit gradient formulas for the Poisson-Szegö kernel, automorphism-based variable changes, and extremal boundary data to establish both the inequality and its sharpness. The authors derive corollaries relating Bergman-metric gradients, hyperbolic Lipschitz continuity, and vector-valued map extensions, and adapt Burgeth's method to obtain a Schwarz-type bound $h(z) \le M_c^n(\|z\|)$ with equality cases described by Poisson integrals of spherical caps. Collectively, the results illuminate how hyperbolic geometry governs invariant harmonic maps on $\mathbb B^n$ and extend classical Schwarz-type phenomena beyond holomorphic functions.

Abstract

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.

Theorems & Definitions (12)

• Theorem A: Liu MR4490949
• Theorem 1
• Corollary 1
• Corollary 2
• Lemma A
• Lemma 1
• proof
• Remark 1
• proof : Proof of Corollary \ref{['cor1']}
• proof : Proof of Corollary \ref{['cor2']}