馮小高

（西華師范大學(xué)數(shù)學(xué)與信息學(xué)院，四川南充637002）

有限偏差函數(shù)與調(diào)和函數(shù)

馮小高

（西華師范大學(xué)數(shù)學(xué)與信息學(xué)院，四川南充637002）

分別借助解析函數(shù)與調(diào)和函數(shù)兩類函數(shù)的Dirichlet積分，利用相關(guān)文獻(xiàn)給定邊界值的擬共形映射極值伸縮商的估計(jì)方法，通過(guò)有限偏差函數(shù)和擬共形映射的關(guān)系估計(jì)了具有給定邊界值的有限偏差函數(shù)的極值伸縮商.得到了解析函數(shù)的Dirichlet積分在有限偏差函數(shù)下具有擬不變性，同時(shí)給出有限偏差函數(shù)極值伸縮商的下界估計(jì).

有限偏差函數(shù)；調(diào)和函數(shù)；Dirichlet積分；極值伸縮商；擬共形映射

1 引言

注1.2由定理1.1可知，解析函數(shù)的Dirichlet積分在有限偏差函數(shù)下具有擬不變性，而對(duì)應(yīng)于定理B中調(diào)和函數(shù)的Dirichlet積分在擬共形映射下具有擬不變性.

2 定理1.1的證明

3 定理1.2的證明

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Mappings of finite distortion and harmonic functions

Feng Xiaogao
（College of Mathmatics and Information，China West Normal University，Nangchong637002，China）

On the basis of Dirichlet integral of holomorphic functions and harmonic functions，rspectively，We shall estimate the extremal maximal dilatation of mappings of finite ditortion with given boundary values by the methods of estimating the extremal maximal dilatation of quasiconformal mappings given boundary values and by using the relation between mappings of finite distortion and quasiconformal mappings.we obtain the quasi-invariance property of the Dirichlet integral under mappings of finite distortion and give the concrete estimates on the extremal dilatation from below.

mapping of finite distortion，harmonic function，holomorphic function，Dirichlet integral，extremal maximal dilatation，quasiconformal mapping

O174

A

1008-5513（2016）02-0119-08

10.3969/j.issn.1008-5513.2016.02.002

2015-10-07.

國(guó)家自然科學(xué)基金（10871211）；西華師范大學(xué)青年教師資助項(xiàng)目（13D017）；西華師范大學(xué)科研基金（08B032）.

馮小高（1982-），博士，講師，研究方向：復(fù)分析.

2010 MSC:30C70，30C62