非齐次多重调和方程拟共形映射的一个Schwarz-Pick型不等式
On a Schwarz-Pick Type Inequality for Quasiconformal Mappings Inhomogeneous Polyharmonic Equation
令?n ∈ C(D),?j ∈ C(T)和K ≥ 1,其中n ≥ 2为整数,j ∈{1,...,n-1}.本文建立了单位圆盘D到自身且满足非齐次多重调和方程Δnf=?n以及相应边界值条件:Δn-1f|T=?n-1,...,Δ1f|T=?1和f(0)=0的K-拟共形映射f的一个Schwarz-Pick型不等式.进一步地,我们证明了这些结果在||?j||∞ → 0(j=1,...,n)和K → 1+的意义下是渐近精确的,其中||?n||∞:=supz∈D|?n(z)|和||?j||∞:=supz∈T|?j(z)|(j=1,2,...,n-1).
Let ?n ∈ C (D), ?j ∈ C (T) and K ≥ 1, where n ≥ 2 is an integer, j ∈ {1,..., n -1}. In this paper, we establish a Schwarz-Pick type inequality for the K-quasiconformal self-mapping f of the unit disk D satisfying the inhomogeneous polyharmonic equation Δnf=?n with the associated Dirichlet boundary value condition:Δn-1f|T=?n-1,..., Δ1f|T=?1 and f(0)=0. Furthermore, we prove that this result is asymptotically sharp in the sense that||?j||∞ → 0 (j=1,..., n) and K → 1+, where||?n||∞:=supz∈D|?n(z)|and||?j||∞:=supz∈T|?j(z)|(j=1, 2,..., n -1).
拟共形性 / 多重调和方程 / Schwarz-Pick型不等式 / 渐近精确 {{custom_keyword}} /
quasiconformality / polyharmonic equation / Schwarz-Pick type inequality / asymptotically sharp {{custom_keyword}} /
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国家自然科学基金(11701111);广东省自然科学基金项目(2016A030310257,2015A030313346);南开大学陈省身数学研究所访问学者计划项目;广东金融学院2020年“优博”科研启动项目(0000-KC2019002001137);广东省普通高校特色创新项目(2019KTSCX111)
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