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

TL;DR

This paper proves a sharp Schwarz-Pick inequality for invariant harmonic functions on the complex unit ball, extending the Khavinson conjecture and revealing that the optimal constants for gradient and radial derivatives are equal.

Contribution

It introduces a new sharp inequality for invariant harmonic functions and solves an extension of the Khavinson conjecture regarding derivative constants.

Findings

Established a sharp Schwarz-Pick inequality for invariant harmonic functions.

Proved that the sharp constants for gradient and radial derivatives are equal.

Derived two corollaries from the main theorem.

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.