A BOUNDARY SCHWARZ LEMMA FOR PLURIHARMONIC MAPPINGS FROM THE UNIT POLYDISK TO THE UNIT BALL

Ling LI; Hongyi LI; Di ZHAO

doi:10.1016/S0252-9602(18)30793-8

Acta mathematica scientia, Series B >

2018 , Vol. 38 >Issue 3: 926 - 934

DOI: https://doi.org/10.1016/S0252-9602(18)30793-8

Articles

A BOUNDARY SCHWARZ LEMMA FOR PLURIHARMONIC MAPPINGS FROM THE UNIT POLYDISK TO THE UNIT BALL

Ling LI ,
Hongyi LI ,
Di ZHAO

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LMIB, School of Mathematics and Systems Science, Beihang University, Beijing 100191, China

Received date: 2017-05-16

Online published: 2018-06-25

Supported by

Supported by the Natural and Science Foundation of China (61379001, 61771001).

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Abstract

In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping f ∈ P(Dⁿ, B^N) is C^1+α at z₀ ∈ E_r ⊂ ∂Dⁿ with f(0)=0 and f(z₀)=w₀ ∈ ∂B^N for any n, N ≥ 1, then there exist a nonnegative vector λ_f=(λ₁,0, …, λ_r, 0, …,0)^T ∈ R²ⁿ satisfying λ_i ≥ 1/2^2n-1 for 1 ≤ i ≤ r such that
(Df(z'₀))^T w'₀=diag(λ_f)z'₀,
where z'₀ and w'₀ are real versions of z₀ and w₀, respectively.

Key words： Boundary Schwarz lemma; pluriharmonic mapping; unit polydisk; unit ball

Cite this article

Ling LI , Hongyi LI , Di ZHAO . A BOUNDARY SCHWARZ LEMMA FOR PLURIHARMONIC MAPPINGS FROM THE UNIT POLYDISK TO THE UNIT BALL[J]. Acta mathematica scientia, Series B, 2018 , 38(3) : 926 -934 . DOI: 10.1016/S0252-9602(18)30793-8

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