Poisson Distribution Results for v-Spirallike Functions of Orderγ

The main objective of this organized paper is to establish the Poisson distribution conditions for the v-spirallike function classes S(γ;ψ)and K(γ;ψ).We also investigate an integral operator associated with the Poisson distribution.

THE INVARIANCE OF STRONG AND ALMOST SPIRALLIKE MAPPINGS OF TYPE β AND ORDER α

The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallike-hess of type β and order α on the unit ball B^n in C^n and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirllikeness of type β, strong and almost starlikeness of order α, strong starlikeness on the corresponding domains.Thus we can construct more subclasses of spirallike mappings in several complex variables.

Homeomorphic extension of strongly spirallike mappings in C^n

In this paper we consider a proper subclass nA of the full class of spirallike mappings on the Euclidean unit ball Bn in Cn with respect to a given linear operator A. We use the method of subordination chains to obtain an upper growth result for nA , and we obtain various examples of mappings in the same class of normalized biholomorphic mappings on the unit ball Bn in Cn . We also prove that the class nA is compact, while the full class of spirallike mappings with respect to a linear operator need not be compact in dimension n≥2, even when the operator is diagonal. This is one of the motivations for considering the class nA . Finally we prove that if f is a quasiregular strongly spirallike mapping on Bn such that ||[Df(z)]-1 Af(z)|| is uniformly bounded on Bn , then f extends to a homeomorphism of R2n onto itself. In addition, if A + A* = 2aI n for some a >0, this extension is also quasiconformal on R2n .

The Growth Theorem and Distortion Theorem of Spirallike Mappings on B_p

In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain Bp. Moreover, the distortion theorem of spirallike mappings is obtained.

THE GENERALIZED ROPER-SUFFRIDGE OPERATOR ON THE UNIT BALL IN COMPLEX BANACH AND HILBERT SPACES

In this paper, we extend the Roper-Suffridge extension operator in complex Ba- nach space, and prove that the extended Roper-Suffridge operators preserve the properties of the subclasses of spirallike mappings on the unit bM1 in complex Banach spaces. Thereby, we promote the conclusions to the cases in complex Hilbert spaces. The conclusions provide new approaches to construct these subclasses of spiraJlike mappings in several complex variables.

PROPERTIES OF THE MODIFIED ROPER-SUFFRIDGE EXTENSION OPERATORS ON REINHARDT DOMAINS

In this paper, we mainly discuss the properties of the modified Roper-Suffridge operators on Reinhardt domains. By the analytical characteristics and distortion results of subclasses of biholomorphic mappings, we conclude that the modified Roper-Suffridge operators preserve the properties of SΩ （β, A, B）, almost starlike mapping of complex order A on Ωn,p2……pn. Sequentially, we get that the modified Roper-Suffridge operators preserve spirallikeness of type β and order a, strongly pirallikeness of type β and order a, almost starlikeness of order a on Ωn,p2……pn The conclusions provide a new approach to construct these biholomorphic mappings which have special geometric properties in several complex variables.

SUFFICIENT CONDITION FOR FUNCTIONS TO BE IN A CLASS OF MEROMORPHIC MULTIVALENT SAKAGUCHI TYPE SPIRAL-LIKE FUNCTIONS

In this article, we define a subclass of meromorphic multivalent Sakaguchi type functions and obtain certain sufficient conditions for functions to be in this class. The main result presented here includes a number of consequences as its special cases.

SUBCLASSES OF BIHOLOMORPHIC MAPPINGS UNDER THE EXTENSION OPERATORS

In this article, we mainly study the invariance of some biholomorphic mappings with special geometric characteristics under the extension operators. First we generalize the Roper-Suffridge extension operators on Bergman-Hartogs domains. Then, by the geometric characteristics of subclasses of biholomorphic mappings, we conclude that the modified Roper-Suff ridge operators preserve the properties of S_Ω~*(β,A, B), parabolic and spirallike mappings of type β and order p, strong and almost spirallike mappings of type 0 and orderα as well as almost starlike mappings of complex order λ on Ω_(p1,…,ps,q)^(B^n) under different conditions, respectively. The conclusions provide new approaches to construct these biholomorphic mappings in several complex variables.

THE INVARIANCE OF SUBCLASSES OF BIHOLOMORPHIC MAPPINGS ON BERGMAN-HARTOGS DOMAINS

We mainly discuss the invariance of some subclasses of biholomorphic mappings under the generalized Roper-Suffridge operators on Bergman-Hartogs domains which are based on the unit ball Bn. Using the geometric properties and the distortion results of subclasses of biholomorphic mappings, we obtain the geometric characters of almost spirallike mappings of type β and order α, S_?~*(β, A, B), strong and almost spirallike mappings of type βand order α maintained under the generalized Roper-Suffridge operators on Bergman-Hartogs domains. Sequentially, we conclude that the generalized operators and the known operators preserve the same properties under some conditions. The conclusions generalize some known results.

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

In this article, we extend the well-known Roper-Suffridge operator on Bn+1 and bounded complete Reinhardt domains in Cn+1, then we investigate the properties of the generalized operators. Applying the Loewner theory, we obtain the mappings constructed by the generalized operators that have parametric representation on Bn+1. In addition, by using the geometric characteristics and the parametric representation of subclasses of spirallike mappings, we conclude that the extended operators preserve the geometric properties of several subclasses of spirallike mappings on Bn+1 and bounded complete Reinhardt domains in Cn+1. The conclusions provide new approaches to construct mappings with special geometric properties in Cn+1.

Extension Operators Preserving Biholomorphic Mappings on Hartogs Domains

In this paper,the authors extend the Roper-Suffridge operator on the generalized Hartogs domains.They mainly research the properties of the extended operator.By the characteristics of Hartogs domains and the geometric properties of subclasses of spirallike mappings,they obtain the extended Roper-Suffridge operator preserving almost starlikeness of complex orderλ,almost spirallikeness of typeβand orderα,parabolic spirallikeness of typeβand orderρon the Hartogs domains in different conditions.They conclude that the corresponding extension operator preserves the same geometric invariance on the unit ball B^(n) in C^(n).The conclusions provide a new approach to study these geometric mappings in C^(n).

The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1|: (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α ＜ 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn: ∑nj=1|zj|pj ＜ 1} (pj ＞ 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α ＜ 1 and some suitable constants βj,γj,pj.

The generalized Roper-Suffridge extension operator in Banach spaces (Ⅲ)

Suppose that X is a complex Banach space with the norm ||·|| and n is a positive integer with dim X ≥n≥ 2. In this paper,we consider the generalized Roper-Suffridge extension operator Φn,β2,γ2,...,βn+1,γn+1(f) on the domain Ωp1,p2,...,pn+1 defined

Area Estimates of Images of Disks under Analytic Functions

Let D_(r):={z=x+iy∈C:|z|<r},r≤1.For a normalized analytic function f in the unit disk D:=D1,estimating the Dirichlet integralΔ(r,f)=∫∫_(D_(r))|f'(z)|^(2) dxdy,z=x+iy,is an important classical problem in complex analysis.Geometrically,Δ(r,f)represents the area of the image of D_(r)under f counting multiplicities.In this paper,our main ob jective is to estimate areas of images of D_(r)under non-vanishing analytic functions of the form(z/f)^(μ),μ>0,in principal powers,when f ranges over certain classes of analytic and univalent functions in D.