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October 12, 2025Open Access

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

Key Points

The Schwarz-Pick lemma was established, outlining specific inequalities for alpha-harmonic mappings.
Explicit forms of sharp functions were derived, showing their dependence on the L^p norms of input functions.
A proven Landau type theorem extends previous theorems to a broader class of alpha-harmonic mappings.
These findings generalize previous results from notable mathematicians, enhancing the field of complex analysis.

Abstract

The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the α-harmonic mapping u=P_αϕ, where ϕ L^p (S^n-1, R) and p1,. We get an explicit form of the sharp function C⏐, ₐ (x) in the inequality | u (x) | C⏐, ₐ (x) \|ϕ\|₋㵵 (ₒ^₍-₁, R). Second, we prove a Landau type theorem for u=P_αϕ, where ϕ L^ (S^n-1, R^n). These results generalize and extend the corresponding results due to Kalaj (Complex Anal. Oper. Theory, 2024) and Khalfallah et al. (Mediterr. J. Math. , 2021).

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

Key Points

Abstract

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