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

        ?

        THE EXTENSION OPERATORS ON Bn+1 AND BOUNDED COMPLETE REINHARDT DOMAINS*

        2020-11-14 09:40:48YanyanCUI崔艷艷ChaojunWANG王朝君
        關鍵詞:劉浩王朝

        Yanyan CUI (崔艷艷)? Chaojun WANG (王朝君)

        College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China

        E-mail : cui9907081@163.com; wang9907081@163.com

        Hao LIU (劉浩)

        Institute of Contemporary Mathematics, Henan University, Kaifeng 475001, China

        E-mail : haoliu@henu.edu.cn

        was introduced in [1], where z =(z1,z0)∈ Bn,z1∈ D,z0=(z2,···,zn)∈ Cn?1,f(z1)∈ H(D)and the branch of the square root is chosen such thatThe operator gives a way of extending a univalent analytic function on the unit disc D in C to a biholomorphic mapping on Bn? Cn. Roper and Suffridge [1]proved that Φn(K) ? K(Bn). Graham and Kohr [2]proved that the Roper-Suffridge operator preserves the properties of Bloch mappings on Bnand Φn(S?)? S?(Bn), and generalized the Roper-Suffridge operator to be

        The above conclusions illustrate that the Roper-Suffridge operator and its extensions are the links between starlike (respectively convex) functions on D and starlike (respectively convex)mappings on Bn. Starlike mappings and convex mappings are important research objects in the geometric function theory of several complex variables. It is easy to find the concrete starlike or convex functions on D, while it is difficult in Cn. By using the Roper-Suffridge operator and its extensions we can construct lots of concrete starlike or convex mappings in Cnfrom the corresponding functions on D. Therefore the Roper-Suffridge operator is useful and important for studying biholomorphic mappings with particular geometric properties in Cn.

        Pfaltzgraff and Suffridge [3]introduced the operator

        on the Euclidean unit ball of Cn, where λj≥ 0 and

        Gong,Liu[4]and Liu,and Liu[5]generalized the Roper-Suffridge operator on more general Reinhardt domains and researched the behaviours of the extension operators. Liu [6]extended the operator (1.1) on bounded starlike circular domains to be

        and proved that the operator(1.2)preserves(almost)starlikeness of order α on bounded starlike circular domains.

        Duan [7]obtained that the generalized operator

        preserves starlikeness on the unit ball Bnin Cn, where Λ = (λij) and Λ is invertible, λij≥ 0,In recent years, there have appeared many results about generalized Roper-Suffridge operators (see [8–11]).

        The above operators all construct locally biholomorphic mappings on different domains in Cnfrom one locally biholomorphic function or n functions f1,··· ,fnin C, while the operator

        introduced by Pfaltzgraff and Suffridge [12], constructs F ∈H(Bn+1) from a locally biholomorphic function f ∈ H(Bn), where n ≥ 1, z′= (z1,··· ,zn) ∈ CnandLetting f(z′)=(f1(z1),··· ,fn(zn)), the operator(1.4)constructs F ∈ H(Bn+1)from n locally biholomorphic functions fi∈ H(D)(i = 1,··· ,n). This stimulates us to extend the Roper-Suffridge operator to be

        where z = (z1,··· ,zn+1) ∈ Bn+1, n ≥ 1, fj(zj) (j = 1,··· ,n) is a normalized locally biholomorphic function on D withfor zj∈ D{0}. For n=1 the operator(1.5)reduces to Φ2,β,as introduced by Graham and Kohr in [2]. The operator (1.5) is not the special form of (1.1) and (1.3); we will shortly investigate the behaviours of (1.5) on Bn+1.

        On the bounded complete Reinhardt domain ? ? Cn+1, we introduce the following extension operator:

        where rj=sup{|zj|:(z1,··· ,zj,··· ,zn+1)′∈ ?}is a normalized locally biholomorphic function on D withfor zj∈D{0}. The operator (1.6) is not the special form of (1.2). Applying (1.5) and (1.6),we can construct biholomorphic mappings in Cn+1(not in Cn) from biholomorphic functions f1,··· ,fnin C.

        In this article, we investigate the properties of the generalized operators(1.5)and(1.6). In section 3, we discuss the fact that the mapping defined by (1.5) has parametric representation on Bn+1if fi(i = 1,··· ,n) does on D, and we research the geometric invariance of some subclasses of spirallike mappings preserved by (1.5) on Bn+1. In section 4, from the geometric characteristics and the parametric representation of subclasses of spirallike mappings,we obtain that (1.6) preserves the geometric properties of several subclasses of spirallike mappings on bounded complete Reinhardt domains in Cn+1.

        2 Definitions and Lemmas

        In what follows, let D denote the unit disk in C, Bndenote the unit ball in Cn, and H(?)denote the holomorphic mappings on ?. Let DF(z) denote the Fréchet derivative of F at z.

        To obtain the main results, we need the following definitions and lemmas:

        Definition 2.1([13]) A mapping f : Bn× [0,∞) → Cnis called a Loewner chain if it satisfies the following conditions:

        (1) f(·,t) is biholomorphic on Bn, f(0,t)=0, Df(0,t)=etIn(t ≥ 0);

        (2) f(z,s)? f(z,t) whenever ? 0 ≤ s ≤ t< ∞ and z ∈ Bn; that is, there exists a Schwarz mapping v =v(z,s,t) such that

        Definition 2.2([13]) Let f(z) be a normalized biholomorphic mapping on Bn. We say that f(z) has parametric representation on Bnif there is a Loewner chain f(z,t) such that{e?tf(·,t)}t≥0is a normal family on Bnand f(z)=f(z,0)(z ∈ Bn).

        Definition 2.3([14]) Let F(z) be a normalized locally biholomorphic mapping on Bn,and α ∈ [0,1), β ∈c ∈ (0,1). Then F(z) is called a strong and almost spirallikemapping of type β and of order α on Bnprovided that

        If we define strong and almost spirallike mappings of type β and of order α on bounded complete Reinhardt domains, the corresponding condition is

        Setting α = 0, β = 0 and α = β = 0, Definition 2.3 reduces to the definition of strong spirallike mappings of type β, strong and almost starlike mappings of order α, and strong starlike mappings, respectively.

        Definition 2.4([15]) Let ? ? Cnbe a bounded starlike circular domain whose Minkowski functional ρ(z)is C1except for a lower-dimensional manifold. Let F(z)be a normalized locally biholomorphic mapping on ?. Then we say thatprovided that

        where ?1 ≤ A

        Setting A = ?1 = ?B ? 2α, A = ?B = ?α, B → 1?in Definition 2.4, we obtain the corresponding definitions of spirallike mappings of type β and order α [16], strongly spirallike mappings of type β and order α [17], and almost spirallike mappings of type β and order α [18]on ?, respectively.

        Definition 2.5([19]) Let ? ? Cnbe a bounded starlike circular domain whose Minkowski functional ρ(z)is C1except for a lower-dimensional manifold. Let F(z)be a normalized locally biholomorphic mapping on ?. Then F(z)is called an almost starlike mapping of complex order λ on ? provided that

        where λ ∈ C,?λ ≤ 0.

        Setting∈ [0,1) in Definition 2.5, we obtain the definition of almost starlikemappings of order α on ?.

        Definition 2.6([20]) Let ? ? Cnbe a bounded starlike circular domain whose Minkowski functional ρ(z)is C1except for a lower-dimensional manifold. Let F(z)be a normalized locally biholomorphic mapping on ? and let ρ ∈ [0,1), β ∈Then F(z) is called a parabolic and spirallike mapping of type β and of order ρ on B provided that

        Suffridge [21]introduced the definition of spirallike mappings with respect to a normal linear operator A whose eigenvalues have a positive real part in complex Banach spaces. Now we extend the definition on bounded complete Reinhardt domains.

        Definition 2.7Let A ∈L(Cn+1,Cn+1) be a continuous complex-linear operator with

        Let ? ? Cn+1be a bounded complete Reinhardt domain and let f :? → Cn+1be a normalized locally biholomorphic mapping. Then f is spirallike relative to A if

        Lemma 2.8([22]) Let f(z) be a normalized locally biholomorphic mapping on B withand a = tan β. Then f(z) is an almost spirallike mapping of order α and of type β on B if and only if g(z,t)=is a Loewner chain.

        Lemma 2.9([13]) Let f(z,t) be a Loewner chain. Then

        Lemma 2.10([23]) Let ? ? Cn+1be a bounded complete Reinhardt domain whose Minkowski functional ρ(z) is C1except for a lower-dimensional manifold. Then we have

        Lemma 2.11([24]) Let ? ? Cnbe a bounded complete Reinhardt domain and let h ∈M, where

        Then the initial value problem

        has a unique solution v(t) = v(z,t) (t ≥ 0). Furthermore, v(z,t)→ 0 (t → +∞) and v(z,t) is a Schwarz mapping on ? for fixed t.

        Lemma 2.12([25]) Let ? ? Cnbe a bounded starlike circular domain and let h ∈M, Jh(0) = A, v(z,t) be a solution of the initial value problem (2.1). Then for ?z ∈ ?,must be existing and converge to a spirallike mapping relative to A on ?. If,instead, f(z) is a spirallike mapping relative to A on ? and Jf(z)h(z)= Af(z), f(z) must be expressed as

        3 Extension Operators on the Unit Ball Bn+1

        In this section we will investigate the properties of mappings constructed by (1.5) on the unit ball Bn+1in Cn+1(not in Cn) from biholomorphic functions f1,··· ,fnin C.

        Theorem 3.1Suppose that the normalized biholomorphic function fk(zk) can be embedded in Loewner chain {gk(zk,t)}t≥0(k = 1,··· ,n) on D. Then the mapping F(z) defined by (1.5) can be embedded in a Loewner chain on Bn+1.

        ProofAs fk(zk)is the normalized biholomorphic function on D,then F(z)is normalized biholomorphic on Bn+1obviously. Because fk(zk) can be embedded in the Loewner chain{gk(zk,t)}t≥0, there exists a Schwarz mapping vk=vk(zk,s,t) such that

        (i) As {gk(zk,t)}t≥0is a Loewner chain, gk(·,t) is biholomorphic on D and gk(0,t) = 0,Therefore F(z,t) is biholomorphic on Bn+1. By a simple calculation we obtain that F(0,t)=0,DF(0,t)=etIn+1where In+1is the identity operator in Cn+1.

        (ii) Let

        Thus W(z,s,t) is a Schwarz mapping on Bn+1.

        (iii) In view of (3.1) and (3.2), we obtain

        From(i)–(iii)and Definition 2.1 we obtain that F(z,t)is a Loewner chain. As fk(zk)can be embedded in the Loewner chain{gk(zk,t)}t≥0,gk(zk,0)=fk(zk),and therefore F(z,0)=F(z).Hence F(z) can be embedded in the Loewner chain F(z,t) on Bn+1.

        Theorem 3.2Suppose that fk(zk)(k =1,··· ,n) is an almost spirallike function of type β and of order α on D with α ∈ [0,1), β ∈Then F(z) defined by (1.5) is an almost spirallike mapping of type β and of order α on Bn+1.

        ProofFrom (1.5) we obtain

        As fk(zk)is an almost spirallike function of type β and of order α on D,gk(zk,t)is the Loewner chain in which fk(zk) is embedded. By (3.3) and (3.4), we obtain

        where F(z,t)is the mapping defined by(3.2)in Theorem 3.1,and thereforeis a Loewner chain. By Lemma 2.8 we obtain that F(z)is an almost spirallike mapping of type β and of order α on Bn+1.

        Theorem 3.3If fk(zk) (k = 1,··· ,n) has parametric representation on D, then F(z)defined by (1.5) has parametric representation on Bn+1.

        ProofAs fk(zk) has parametric representation on D, there exists a Loewner chain gk(zk,t) such that gk(zk,0) = fk(zk), and {e?tgk(·,t)}t≥0is a normal family. By Lemma 2.9 we obtain

        Therefore, for |zk|

        By Theorem 3.1 we obtain that F(z,t)defined by(3.1)is a Loewner chain,that F(z,0)=F(z)and

        Therefore {e?tF(·,t)}t≥0is locally uniformly bounded on Bn+1, and thus is a normal family.By Definition 2.2 we obtain that F(z) has parametric representation on Bn+1.

        Theorem 3.4Suppose that fk(zk) (k =1,··· ,n) is a strong and almost spirallike function of type β and of order α on D with α ∈ [0,1) and β ∈Then F(z) defined by(1.5) is a strong and almost spirallike mapping of type β and of order α on Bn+1.

        ProofBy (1.5) we obtain

        As fj(zj) (j =1,··· ,n) is normalized locally biholomorphic on D withandis normalized locally biholomorphic on Bn+1and

        By Definition 2.3, we need only to prove that

        Then |qk(zk)|<1. In view of c ∈(0,1) and (3.5), we obtain

        Therefore F(z) is a strong and almost spirallike mapping of type β and of order α on Bn+1by Definition 2.3.

        Similarly to Theorem 3.4 we can obtain the following conclusion:

        Theorem 3.5Let(k = 1,··· ,n) with ?1 ≤A < B < 1 andLetting F(z)be the mapping defined by(1.5),we get that F(z)∈

        Remark 3.6Setting β = 0 (respectively α = 0) in Theorem 3.4, we obtain the corresponding results for strong and almost starlike mappings of order α (and, respectively,the strongly spirallike mappings of type β). Setting A = ?1 = ?B ? 2α (respectively,A = ?B = ?α) in Theorem 3.5, we obtain the corresponding results for spirallike mappings of type β and of order α (and, respectively, the strongly spirallike mappings of type β and of order α).

        Theorem 3.7Let fk(zk) (k =1,··· ,n) be an almost starlike function of complex order λ on D with λ ∈ C,?λ ≤ 0. Then F(z) defined by (1.5) is an almost starlike mapping of complex order λ on Bn+1.

        ProofBy Definition 2.5, we need only to prove that

        As fk(zk) is an almost starlike function of complex order λ on D, by Definition 2.5, we have that

        Then ?pk(zk)≥ 0. By (3.5) we obtain that

        In addition, ?pk(zk) ≥0 implies thatTherefore F(w,z) is an almost starlike mapping of complex order λ on Bn+1.

        Remark 3.8Settingin Theorem 3.7, we obtain the corresponding results for almost starlike mappings of order α.

        4 Extension Operators on Bounded Complete Reinhardt Domains

        Let ? ? Cn+1be a bounded starlike circular domain whose Minkowski functional ρ(z) is C1except for a lower-dimensional manifold. In this section we will study the properties of(1.6), preserving several subclasses of spirallike mappings on ?.

        Theorem 4.1Let(k = 1,··· ,n) with ?1 ≤A < B < 1 andLet F(z) be the mapping defined by (1.6). Then

        ProofBy Definition 2.4, we need only to prove that

        Then |qk(ξk)|<1. By (4.2) we obtain that

        which follows (4.1). By Definition 2.4 we obtain that

        Similarly, we can obtain that F(z) defined by (1.6) preserves the strongth and almost spirallikeness of type β and of order α on ?.

        Theorem 4.2Let(k = 1,··· ,n) be a strong and almost spirallike function of type β and of order α on D with α ∈ [0,1) andThen F(z) defined by (1.6) is a strong and almost spirallike mapping of type β and of order α on ?.

        Remark 4.3Setting A= ?1= ?B ?2α (respectively, A= ?B = ?α) in Theorem 4.1,we obtain the corresponding results for spirallike mappings of type β and of order α (respectively,the strongly spirallike mappings of type β and of order α). Setting β =0 (respectively, α =0)in Theorem 4.2, we obtain the corresponding results for strong and almost starlike mappings of order α (and respectively, the strongly spirallike mappings of type β).

        Theorem 4.4Let(k =1,··· ,n) be a parabolic and spirallike function of type β and of order ρ on D with ρ ∈ [0,1), β ∈and ρ ≤ cos β. Then F(z) defined by (1.6) is a parabolic and spirallike mapping of type β and of order ρ on ?.

        ProofBy Definition 2.6, we need only to prove that

        As fk(ξk) is a parabolic and spirallike function of type β and of order ρ on D, we have that

        By (4.2) we obtain that

        In view of ρ ≤ cos β, by (4.4) we obtain that

        which follows(4.3). Therefore F(z)is a parabolic and spirallike mapping of type β and of order ρ on ?.

        Remark 4.5Setting ρ = 0 (respectively, β = 0) in Theorem 4.4, we obtain the corresponding results for parabolic and spirallike mappings of type β (respectively, the parabolic and starlike mappings of order ρ).

        Similarly to Theorem 4.4, we can obtain the following conclusion:

        Theorem 4.6Let(k =1,··· ,n) be an almost starlike function of complex order λ on D with λ ∈ C,?λ ≤ 0. Then F(z) defined by (1.6) is an almost starlike mapping of complex order λ on ?.

        For rj=1 (j =1,··· ,n), (1.6)reduces to(1.5). Therefore,from the above conclusions,we can obtain that (1.5) preserves the properties of the following biholomorphic mappings on the Reinhardt domain:

        Corollary 4.7Let fk(zk)be a strong and almost spirallike function of type β and of order α on D (respectively,a parabolic and spirallike function of type β and of order ρ,and an almost starlike function of complex order λ). Then F(z)defined by(1.5)is a strong and almost spirallike mapping of type β and of order α on ?n+1,p(respectively,a parabolic and spirallike mapping of type β and of order ρ, and an almost starlike mapping of complex order λ).

        Remark 4.8For pn+1= 2, ?n+1,preduces to Bn+1. Therefore, by corollary 4.7 we obtain the corresponding results for (1.5) on Bn+1.

        In what follows we will research the properties of the operator(1.6)on the bounded complete Reinhardt domain ? from the parametric representation of spirallike mappings.

        Theorem 4.9Let(k =1,··· ,n)be a spirallike function relative to A on D with A being a continuous complex-linear operator. Then F(z)defined by(1.6)is a spirallike mapping relative to A on ?, where

        Proof(i) Asis a spirallike function relative to A on D, there existssuch that

        Let F(z)=(F1(z),··· ,Fn+1(z)). For z ∈ ? we have that

        By (4.7) and (4.9), we obtain that?H(u(z,t)), where

        Furthermore,

        that is,u(z,0)=z. Therefore u(z,t)is the solution of the initial value problem(2.1)in Lemma 2.11.

        (iii) In view of hj∈M and (4.8), we have that

        Applying Lemma 2.10, we obtain that

        so H ∈M.

        By (4.5) we have that

        By simple calculation, we obtain that

        For H(u(z,t))=(u1q1(u),··· ,un+1qn+1(u))′, let Hj(u)=ujqj(u) (j =1,··· ,n+1). By (4.8)we get thatfor j =1,··· ,n, therefore

        By (4.9) we get that

        By (4.11)–(4.13) we obtain that JH(0)=A.

        From (i)–(iii) and lemma 2.12 we obtain that F(z) is a spirallike mapping relative to A on?.

        Corollary 4.10Let(k = 1,··· ,n) be a spirallike function relative to A on D.Then

        is a spirallike mapping relative to A on ?, where

        Remark 4.11Setting A=e?iβIinto Theorem 4.9 and Corollary 4.10, we obtain the corresponding results for spirallike mappings of type β.

        猜你喜歡
        劉浩王朝
        正確看待輸和贏
        多重映射芽的Gq,k一決定性
        進球了
        Negative compressibility property in hinging open-cell Kelvin structure*
        養(yǎng)心殿,帶你走進大清王朝的興衰沉浮
        金橋(2018年10期)2018-10-09 07:27:44
        奇波利尼的王朝Saeco
        中國自行車(2018年8期)2018-09-26 06:53:08
        PROPERTIES OF THE MODIFIED ROPER-SUFFRIDGE EXTENSION OPERATORS ON REINHARDT DOMAINS?
        劉浩藝術作品欣賞
        消除“鈍”感肌就是這樣滑!
        Coco薇(2015年3期)2015-12-24 03:06:17
        THE INVARIANCE OF STRONG AND ALMOSTSPIRALLIKE MAPPINGS OF TYPE β AND ORDER α?
        日本午夜国产精彩| 精品国产免费一区二区三区| 男人激烈吮乳吃奶视频免费| 欧美激情在线不卡视频网站| 淫欲一区二区中文字幕| 六月婷婷亚洲性色av蜜桃| 久久久www成人免费毛片| 亚洲暴爽av天天爽日日碰| 少妇bbwbbw高潮| 国产乱子伦一区二区三区国色天香| 国产精品福利一区二区| 国产95在线 | 欧美| 亚洲欧洲日产国码无码| 国产人妖伦理视频在线观看| 人妻洗澡被强公日日澡电影| 两个人看的www高清视频中文| 中文字幕乱码人妻无码久久久1 | 亚洲av无码av吞精久久| 国产AV高清精品久久| 精华国产一区二区三区| 亚洲av片在线观看| 亚洲精品免费专区| 成av人片一区二区三区久久| 少妇一级淫片中文字幕| 久久久精品456亚洲影院| 久久99精品中文字幕在| 成人国产高清av一区二区三区| 久久精品国产精品亚洲| 精品国产一区二区三区久久久狼 | 亚洲中文字幕无线无码毛片| 国产免费99久久精品| 亚洲国产亚综合在线区| 亚州少妇无套内射激情视频| 国产桃色精品网站| 五月婷婷六月丁香久久综合| 亚洲国产成人片在线观看无码| 午夜亚洲AV成人无码国产| 日本在线观看三级视频| 艳妇臀荡乳欲伦69调教视频| 男女真实有遮挡xx00动态图| 国产精品一级黄色大片|