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

        ?

        DISTORTION THEOREMS FOR CLASSES OFg-PARAMETRIC STARLIKE MAPPINGS OF REAL ORDER IN Cn*

        2024-01-12 13:17:36HongyanLIU劉紅炎ZhenhanTU涂振漢SchoolofMathematicsandStatisticsWuhanUniversityWuhan430072Chinamailhongyanliuwhueducnzhhtumathwhueducn

        Hongyan LIU(劉紅炎)Zhenhan TU(涂振漢)School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China E-mail: hongyanliu@whu.edu.cn; zhhtu.math@whu.edu.cn

        Liangpeng XIONG (熊良鵬)i School of Mathematics and Computer Science,Jiangri Science and Technology Normal Universitgy,Nanchang 330038,China E-mail: lpxionq2016@whu.edu.cn

        In the geometric theory of one complex variable, the following distortion theorem for biholomorphic functions is well known:

        Theorem A (Duren [5]) Let f be a biholomorphic function on the unit disk D = {z ∈C:|z|<1}, and let f(0)=f′(0)-1=0.Then

        However, for the case of several complex variables, Cartan [2] has pointed out that the above theorem does not hold.Thus,it is necessary to study the properties for some special subclasses of biholomorphic mappings defined on different domains, for example, starlike mappings, convex mappings, close-to-convex mappings, Bloch mappings, and so on (see, e.g., Chu-Hamada-Honda-Kohr [4], Graham-Hamada-Kohr [9], Hamada-Honda-Kohr [10], Liu [15], Wang [20]).Barnard-FitzGerald-Gong [1] first studied the distortion theorems of Jacobi-determinant type for convex mappings defined on the Euclidean unit ball, and subsequently, there appeared a lot of important results regarding distortion theorems for convex mappings(see,e.g.,Chu-Hamada-Honda-Kohr [3], Gong-Wang-Yu [6], Gong-Liu [7], Xu-Liu [22], Zhu-Liu [24]).Compared with the case of convex mappings, we know that the geometric properties of biholomorphic starlike mappings are weaker, and thus it is much more difficult to obtain the corresponding distortion theorems for certain kinds of subclasses of starlike mappings.For the space of n-dimensional(n ≥2) complex variables, Poreda[17] discussed the biholomorphic mappings of the unit polydisk in Cnwhich have a parametric representation, After that, some further results regarding subclasses of biholomorphic mappings which have a parametric representation were obtained with different targets (see, e.g., Kohr-Liczberski[11], Kohr [12], Tu-Xiong [19]).Recently, Liu-Liu[14]discussed the distortion theorems with respect to a subclass of biholomorphic mappings which have a parametric representation in several complex variables (also see, e.g., Graham-Hamada-Kohr [8], Xiong [23] ).This paper will make further progress along these lines.

        First, we introduce some notations and definitions.

        Let A denote the class of all analytic functions on the unit disk D = {z : |z| < 1} in C.Denote by S the subclass of A consisting of functions that are univalent.If f and g are analytic in D, we say that f is subordinate to g,writing this as f ?g, provided that there exist analytic functions ω(z) defined on D with ω(0)=0 and |ω(z)|<1 satisfying that f(z)=g(ω(z)).

        A function f ∈A is said to belong to the class S?(γ) of starlike functions of complex order γ if it satisfies the following inequality:

        The function class S?(γ) was considered by Nasr-Aouf [16] (also see Srivastava et al., [18]).In particular, this is the usual class of starlike functions in D when γ = 1.The following natural questions arise regarding the dimensions when n ≥2:

        Question 1.1 Can we extend the definition of the class S?(γ) from the case in onedimensional space to the case in n-dimensional space (n ≥2)?

        Question 1.2 Can we establish the distortion theorems of the class of starlike mappings of complex order γ in the n-dimensional complex variables space (n ≥2)?

        We shall try to give affirmative answers to the above questions.

        Let g ∈H(D) be a univalent function such that g(0)=1,g(ζ)=g(ζ) for ζ ∈D (i.e., g has real coefficients) and ?g(ζ) > 0 on D, and assume that g satisfies the following conditions for r ∈(0,1) (see Xu-Liu [22]):

        We define the following class of g-parametric starlike mappings of real order γ on BX(γ ∈(0,1]),this is closely related to the g-parametric starlike mappings on BX:

        Definition 1.3 Let f ∈H(BX),f(0) = 0,Df(0) = I,0 < γ ≤1 and let the function g satisfy the condition (1.2).Then

        where x ∈BX{0}, Tx∈T(x).

        Remark 1.4 (i) If we choose the parametric γ ∈C?in Definition 1.3, then the existing critical Lemma 2.1 looks like failure.Thus, we cannot make sure that the main theorems hold true in this situation.

        (ii) If we take γ =1 and replace the function g by G in Definition 1.3, where G=1g, then it reduces to the class defined by Liu-Liu [14].

        (iii) Let Cnbe the space of n-dimensional complex variables z =(z1,z2,··· ,zn) with the maximum norm‖z‖=‖z‖∞=max{|z1|,|z2|,··· ,|zn|}.Denote by Dnthe unit polydisc in Cn.Therefore, if we take X=Cnand BX=Dnin Definition 1.3, then

        where 0 < |zj| = ‖z‖∞< 1, h(z) = [Df(z)]-1f(z) = (h1(z),h2(z),··· ,hn(z)) and g satisfies the condition(1.2).As usual,we write a point z ∈Cnas a column vector in the following n×1 matrix form:

        The derivative of f ∈H(Dn) at a point a ∈Dnis the complex Jacobian matrix of f given by

        We denote by det Jf(z) the Jacobi determinant of the holomorphic mapping f.

        2 Preliminaries

        In this section,we give some lemmas which play a key role in the proof of our main theorems.

        In (2.2), setting ζ =‖x‖ yields the desired inequalities.□

        3 Main Results

        We derive the desired results from (3.21) and (3.24).□

        4 Some Corollories

        In this section, we give some corollaries by using Theorems 3.1 to 3.4, which are the corresponding distortion results for subclasses of g-starlike mappings defined on B (resp.Dn).If we replace the function g by G in Corollaries 4.1 to 4.4, where G =1g, then we see that these results were proven by Liu-Liu [14].

        Corollary 4.1 Suppose that the function g satisfies the condition (1.2) and that fl:Dml→C are some holomorphic functions, l=1,2,··· ,n.Let

        where α ∈[0,1), β ∈(0,1], c ∈(0,1), ξ ∈D.It is easy to verify that every function g conforms to the condition (1.2) whenever g ∈M.Therefore, if we take a certain function g ∈M in Theorems 3.1 to 3.4, then the distortion theorems for certain kinds of subclasses of biholomorphic g-starlike mappings of real order γ defined on Dn(resp.B) can be obtained immediately.

        Conflict of InterestThe authors declare no conflict of interest.

        欧美乱人伦人妻中文字幕| 日日麻批视频免费播放器| 在线观看在线观看一区二区三区| 久久黄色视频| 先锋影音av最新资源| 亚洲国产99精品国自产拍| 亚洲国产成人精品一区刚刚 | 亚洲写真成人午夜亚洲美女| 精品久久久久久亚洲综合网| 少妇装睡让我滑了进去| 久久精品国产亚洲婷婷| 永久免费看黄在线观看| 综合偷自拍亚洲乱中文字幕 | 亚洲a∨无码一区二区| 国产九色AV刺激露脸对白 | 亚洲精品国产av日韩专区| 老妇高潮潮喷到猛进猛出| 99精品免费久久久久久久久日本| 亚洲不卡电影| 亚洲女人毛茸茸的视频| 亚洲欧洲成人a∨在线观看| 亚洲 欧美 激情 小说 另类| 东京热无码人妻中文字幕| 中文字幕一区二区综合| 免费va国产高清大片在线| 免费一区在线观看| 亚洲午夜精品国产一区二区三区| 日韩一区在线精品视频| 四虎国产精品免费久久| 亚洲欧美日韩国产精品网| 国产诱惑人的视频在线观看| 日本少妇春药特殊按摩3| 精品国精品国产自在久国产应用| 人妻中文字幕一区二区二区| 国产av剧情刺激对白| 亚洲精品无码久久久久牙蜜区| 亚洲AV激情一区二区二三区| 亚洲精品综合久久中文字幕| 欧美最猛黑人xxxx| 国产一级农村无码| 妺妺窝人体色www在线图片 |