亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        A Generalized Schwarz Lemma

        2017-06-05 15:01:17ZHAOLiankunLIHongyi

        ZHAO Lian-kun,LI Hong-yi

        (LIIB.School of Mathematics and System Science,Beihang University,Beijing 100191,China)

        A Generalized Schwarz Lemma

        ZHAO Lian-kun,LI Hong-yi

        (LIIB.School of Mathematics and System Science,Beihang University,Beijing 100191,China)

        In this paper,we establish a boundary Schwarz Lemma for holomorphic mapping on the generalized complex ellipsoid in Cn.

        several complex variables;Schwarz Lemma;generalized complex ellipsoid

        §1.Introduction

        Schwarz lemma is the most fundamental and significant result in complex analysis,having attracted wide attention and analysis over the past century.This noted lemma reveals that the range of homomorphic functions,which satisfy certain condition,will be compressed along with its domain of definition.As is well known,Riemann mapping theorem is no longer valid in several complex variables.Even two of the most simple domains unit ball Bnand polydisc Dnin several complex variables are not holomorphic equivalent.Therefore,it is essential to study the corresponding versions of Schwarz lemma in distinct domains in several complex variables.

        1.1 Schwarz Lemma in One Complex Variable

        In one complex variables,the following Schwarz Lemma is classical[1].

        Theorem ALet is unit disk in complex plane C and f:D→D is a homomorphic function.If f(0)=0,then?ζ∈D,

        The above equality holds if and only if f(z)=cz,where|c|=1.

        The classical result was first introduced by H A Schwarz in 1869.According to this lemma, if a holomorphic function f of D into itself with f(0)=0,then the module of range can be controlled by the module of domain of definition.

        Given a boundary point,there is a boundary Schwarz Lemma

        Theorem BLet is a unit disk in complex plane C and let f:D→D be a holomorphic function.Iffis holomorphic at z=1 and f(0)=0,f(1)=1,then f′(1)≥1.Moreover,the inequality is sharp.

        RemarkNotice that,if the condition of f(0)=0 in Theorem A is removed and apply Theorem B to

        one can get

        Before taking a closer look at Schwarz lemma,some notations and definitions are given.

        1.2 Some Notations and Definitions

        The derivative of f=(f1,f2,···,fn)Tat z0is the complex Jacobi matrix of f,which is defined as follows,

        Definite 1Let ? be a domain in Cn.If for any(z1,···,zn)T∈? and θ1,···,θn∈R, one can get

        then ? is called Reinhardt domain.

        Definite 2Generalized complex ellipsoid in Cnis defined as follows,

        where n≥2,p>0.

        Definite 3z0∈Bp,n,the tangent space Tz0(?Rp)to?Rpat z0is defined by

        Notice that,generalized complex ellipsoid Bp,nis bounded and is a special kind of Reinhardt domain.

        1.3 Relative Results

        Theorem 1[2]Let ? denote bounded domains in Cn,f is a holomorphic function of ?, p∈? is a fixed point,then

        (1)The module of all eigenvalues of Jf(p)less than 1;

        (2)|det Jf(p)|≤1;

        (3)If|det Jf(p)|=1,then f is a biholomorphism mapping of ?.

        Theorem 2[3]Let ? be a bounded strongly pseudoconvex domain in Cn,p∈??,f:?→? is a holomorphic function,when z→p,f(z)=z+O(|z-p|4),then f(z)≡z.

        Theorem 3[4]Let ? be a strictly convex domains in Cn(n>1),p∈??,f:?→?is a holomorphic function,for a certain point z0∈?,f(z0)=z0and when z→p,f(z)= z+O(|z-p|2).then f(z)≡z.

        For the past few years,plenty of research and development work is being done about Schwarz Lemma in many distinctive domains in Cn,which includes boundary Schwarz Lemma between unit ball Bnin Cn[5],between polydisk Dnin Cn[6],form polydisk Dnto unit ball Bn[7], strictly convex domains Cn[8],strongly pseudoconvex domain[9]and between a special kind of Reinhardt domain Bp1,p2={(z1,z2,···,zn)T∈Cn:|z1|p1+|z2|p2+···+|zn|p2<1}, p1≥1,p2≥1[10].

        §2.A Boundary Schwarz Lemma for Holomorphic Mapping on the Generalized Complex Ellipsoid in Cn

        We establish a boundary Schwarz Lemma of generalized complex ellipsoid Bp,nin this paper and the main results are as follows.

        Theorem 2.1Let Bp,nis a generalized complex ellipsoid in Cn,where n≥2,p>0. f:Bp,n→Bp,nis a holomorphic mapping,z0∈S,where S=?Bp,nT?Bn,f is holomorphic at z0,f(z0)=z0,then the eigenvalues of Jf(z0)λ1,μ2,···,μnsatisfy the following statements

        (1)z0is the eigenvector of Jf(z0)Hwith respect to λ.That is to say,Jf(z0)Hz0=λz0;

        Proof

        So,for any α∈Tz0(?Bp,n),

        This shows that λ is a eigenvector of Jf(z0)H.λ is a real number,then λ is the eigenvalue of Jf(z0)

        then U is a unitary matrix of order n.Because Jf(z0)Hz0=λz0,we get

        where V is a complex matrix of order(n-1),B is an complex matrix of order(n-1)×1. Hence,

        that is,

        the roots of det(xIn?1-V)areμ2,···,μn.If λ/∈{μ2,···,μn},then λ,μ2,···,μnare all eigenvalues of Jfz0.

        Suppose that λ=μi0is a zero point of order k of det(xIn?1-V)and Jf(z0)is a linear transformation on Cn,then characteristic polynomial of Jf(z0)is

        (3)According to hypothesis S=?Bp,nT?Bn,obviously(eiθ,0,···,0)T,···,(0,···,0,eiθ)T∈S,so S is not empty.

        Without loss of generality,z0=(eiθ,0,···,0)T.z0=(0,···,0,eiθ,0,···,0)T,j-th element of which is eiθ,can be transformed to z0=(eiθ,0,···,0)Tby a simple permutation transformation ψ

        where a=f(0),if a=0,it follows that λ≥1.

        [1]YIN Wei-ping.The Bergman kernel function of super-Cartan domain of the first type[J].Science in China, 1999,29A(7):607-615.

        [1]GARNETT J B.Bounded Analytic Functions[M].New York:Springer,2007.

        [2]WU Hong-xi.Normal families of holomorphic mappings[J].Acta Mathematica,1967,119(1):193-233.

        [3]BURNS D M,KRANTZ S G.Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary[J].Journal of the American Mathematical Society,1994,7(3):661-676.

        [4]HUANG Xiao-jun.A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains[J].Canadian Journal of Mathematics,1995,47(2):405-420.

        [5]LIU Tai-shun,WANG Jian-fei,TANG Xiao-min.Schwarz Lemma at the boundary of the unit ball in C n and its applications[J].The Journal of Geometric Analysis,2015,25(3):1890-1914.

        [6]TANG Xiao-min,LIU Tai-shun,LU Jin.Schwarz Lemma at the boundary of the unit polydisk in Cn[J]. Science China Mathematics,2015,58(8):1639-1652.

        [7]LIU Yang,CHEN Zhi-hua,PAN Yi-fei.A boundary Schwarz Lemma for holomorphic mappings from the polydisc to the unit ball[J].2014,arXiv preprint arXiv:1411.0603.

        [8]WANG Xie-ping.Schwarz Lemma at the boundary of strongly convex domains in Cn[J].Complex Analysis and Operator Theory,2015,11(2):1-14.

        [9]LIU Tai-shun,TANG Xiao-min.Schwarz Lemma at the boundary of strongly pseudoconvex domain in Cn[J].Mathematische Annalen,2016,366(1-2):655-666.

        [10]ZHANG Xiao-fei.Schwarz Lemma at the Boundary for Several Complex Variables and Its Applications[D]. Hefei:University of Science and Technology of China,2013.

        tion:32H02,30C80,32A30

        :A

        1002–0462(2017)01–0001–06

        date:2016-10-23

        Supported by the National Natural Science Foundation of China(61379001)

        Biographies:ZHAO Lian-kun(1991-),male,native of Tengzhou,Shandong,a master of mathematics of Beihang University,engages in complex analysis;LI Hong-yi(corresponding author)(1960-),female,native of Kaifeng,Henan,a professor of Beihang University,Ph.D.,engages in complex analysis and its application.

        CLC number:O174.56

        日韩精品首页在线观看| 麻豆精品久久久久久久99蜜桃| 亚洲欧美综合在线天堂| 韩国日本亚洲精品视频 | 亚洲区日韩精品中文字幕| 一区二区三区在线免费av| 久久精品人搡人妻人少妇| 欧美大成色www永久网站婷| 亚洲综合色一区二区三区另类| 精品女同一区二区三区不卡| 国产变态av一区二区三区调教 | 国产乱妇乱子视频在播放| 免费在线观看一区二区| 成人爽a毛片在线播放| 精品乱码一区内射人妻无码 | 天天干夜夜躁| 东京热加勒比国产精品| 三年片在线观看免费观看大全中国| 欧美日本国产va高清cabal| 狠狠综合亚洲综合亚色| 亚洲精品女同一区二区三区| 日韩一区国产二区欧美三区| 精品国产免费一区二区三区| 极品美女扒开粉嫩小泬| 中文字幕一区二区人妻痴汉电车| 国产性虐视频在线观看| 女人被狂c躁到高潮视频| 亚洲人妻无缓冲av不卡| 在线久草视频免费播放 | 免费少妇a级毛片人成网| 天天躁日日操狠狠操欧美老妇| 亚洲精品中文字幕熟女| 精品国品一二三产品区别在线观看| 人妻无码人妻有码中文字幕| 激情人妻中出中文字幕一区| 亚洲一区二区三区特色视频| 免费无码黄动漫在线观看| 国产片三级视频播放| 粉嫩人妻91精品视色在线看| 全免费a敌肛交毛片免费| 国产爽爽视频在线|