Schwarz-Pick type lemma and Landau type theorem for α-harmonic mappings

2025/09/08 by Vibhuti Arora, Arora, Vibhuti, Jiaolong Chen +5
Mathematics · Computer Science · #Analytic and geometric function theory #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.06359

Abstract

The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the α-harmonic mapping u=P_α[ϕ], where ϕ∈ L^p(\mathbbS^n-1,ℝ ) and p∈[1,∞]. We get an explicit form of the sharp function C_{α, q}(x) in the inequality |∇ u(x)| ≤ C_{α, q}(x)‖ϕ‖_L^p(\mathbbS^n-1, ℝ ). Second, we prove a Landau type theorem for u=P_α[ϕ], where ϕ∈ L^∞(\mathbbS^n-1,ℝⁿ). These results generalize and extend the corresponding results due to Kalaj (Complex Anal. Oper. Theory, 2024) and Khalfallah et al. (Mediterr. J. Math., 2021).

Citations

• Some optimal inequalities for alpha-harmonic functions estimated by their boundary functions
• Schwarz lemma for the solutions to the Dirichlet problems for the invariant Laplacians
• Schwarz Lemma and Schwarz-Pick Lemma for solutions of the α-harmonic equation
• The Landau-Bloch type theorems for certain class of holomorphic and pluriharmonic mappings in \mathbbcⁿ
• Isoperimetric type inequalities for mappings induced by weighted Laplace differential operators
• Schwarz-Pick Lemma for Harmonic and Hyperbolic Harmonic Functions
• Schwarz-Pick and Landau type theorems for solutions to the Dirichlet-Neumann problem in the unit disk
• Schwarz-Pick lemma for harmonic functions
• Weighted integrability of polyharmonic functions in the higher dimensional case
• Solution to the Khavinson problem near the boundary of the unit ball
• Some properties of a class of elliptic partial differential operators
• Harmonic functions on the real hyperbolic ball I : Boundary values and atomic decomposition of Hardy spaces

Related

• Landau type theorem for α-harmonic mappings
• Several properties of α-harmonic functions in the unit disk
• Sharp pointwise estimate of α-harmonic functions
• Schwarz Lemma and Schwarz-Pick Lemma for solutions of the α-harmonic equation
• Weighted Lipschitz continuity, Schwarz-Pick's Lemma and Landau-Bloch's theorem for hyperbolic-harmonic mappings in ℂⁿ