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.

💡 Research Summary

The paper investigates real‑valued invariant harmonic functions on the complex unit ball 𝔹ⁿ = {z ∈ ℂⁿ : ‖z‖ < 1}. An invariant harmonic function h satisfies the Laplace–Beltrami equation Δ_B h = 0 with respect to the Bergman metric g, which is the natural hyperbolic metric on 𝔹ⁿ. The authors prove a sharp Schwarz‑Pick type gradient estimate:

  ‖∇h(z)‖ ≤ 2 Γ(n + 1) √π Γ(n + ½) · (1 − ‖z‖²)⁻¹,  z ∈ 𝔹ⁿ,

and show that the constant is optimal. The proof proceeds by representing h as a Poisson‑Szegő integral of its boundary values, differentiating under the integral sign, and estimating the directional derivative ∂h/∂l(z) for an arbitrary unit vector l ∈ ∂𝔹ⁿ. By applying an automorphism ϕ_z of the ball that sends z to the origin, the authors reduce the problem to an integral over the sphere that is independent of l up to a factor ‖v(z,l)‖ ≤ 1, where v(z,l) = s_z l + (1 − s_z)⟨l,z⟩z/‖z‖² and s_z = √(1 − ‖z‖²). Explicit evaluation of the spherical integral using beta and gamma functions yields the constant displayed above.

Sharpness is demonstrated by choosing boundary data equal to the sign of the kernel’s gradient, which produces an extremal invariant harmonic function attaining equality for every point z. In particular, when l is taken in the radial direction (l = z/‖z‖), the factor ‖v(z,l)‖ equals 1, confirming that the maximal directional derivative occurs radially. This resolves the analogue of the Khavinson conjecture for invariant harmonic functions: the optimal constants for the radial derivative and the full gradient coincide.

Two corollaries follow. First, by comparing the Euclidean gradient with the Bergman‑gradient ∇B, the authors obtain a Lipschitz estimate with respect to the hyperbolic distance d{𝔹ⁿ}:

  |h(z) − h(w)| ≤ C_n d_{𝔹ⁿ}(z,w),

where C_n = 2 Γ(n + 1) π^{1/2} Γ(n + ½). This shows that bounded invariant harmonic functions are distance‑decreasing in the hyperbolic metric, mirroring the classical Schwarz‑Pick lemma.

Second, for vector‑valued maps H = (h₁,…,h_m) whose components are invariant harmonic, the same constant controls the operator norm of the Jacobian:

  ‖∇H(z)‖ ≤ 2 Γ(n + 1) √π Γ(n + ½) · (1 − ‖z‖²)⁻¹.

Thus the scalar estimate extends to the matrix‑valued setting without loss of sharpness.

In the final section the authors adapt Burgeth’s method to obtain a Schwarz‑type bound for invariant harmonic functions with prescribed value at the origin. For h(0)=a (−1<a<1) they define c = (a+1)/2 and a spherical cap S(c,ẑ) on the boundary. The extremal function is the Poisson integral of the characteristic function of this cap, yielding an explicit upper bound

  h(z) ≤ M_{n,c}(‖z‖) = 2∫{∂𝔹ⁿ}1{S(c,ẑ)}(w) P_z(w) dσ(w) − 1.

In dimension n=1 this reduces to the classical formula M_{1,c}(r)=4π arctan