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 spirallikeness 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 spirallikeness 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.     

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 S_Ω~*(β,A,B),almost starlike mapping of complex orderλ on Ω_(n,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 α,almost starlikeness of order α on Ω_(n,p2,…,pn).The conclusions provide a new approach to construct these biholomorphic mappings which have special geometric properties 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.     

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 order λ, almost spirallikeness of type β and order α, parabolic spirallikeness of type β and order ρ on the Hartogs domains in different co...     

Properties of New Holomorphic Mappings with Respect to Conic Domains

This paper is mainly about holomorphic mappings associated with conic regions which are closely connected with k-ST(α). We introduce new subclasses of starlike(spirallike) functions,namely, S_c~p(k, α)(S_c~p(k, α, β)), and discuss their coefficient estimates and the Fekete–Szeg?–Goluzin's problem. Then we generalize S_c~p(k, α, β) on the unit ball B~n in C~n, that is, k-conic spirallike mappings of type β and order α. We obtain the growth, covering and distortion theorems of the generalized mappings. Besides that, we construct k-conic spirallike mappings of type β and order α on B~n through S_c(k, α, β) by the generalized Roper-Suffridge extension operators.     

On Growth and Covering Theorem for Almost Spiallike Mapping of Type β with Order α

Let f(x) be an almost spirallike mapping of type β with order α on unit ball B of complex Banach space X. In this paper, we consider the growth and covering theorems for f(x), we also prove that the estimation is precise when β=0 and still give growth upper bound and distortion upper bound for subordinate mapping. This result include some results known.     

SHARP ESTIMATES OF ALL HOMOGENEOUS EXPANSIONS FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS OF TYPE B AND ORDER α IN SEVERAL COMPLEX VARIABLES

In this article,first,the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order α on the unit ball in complex Banach spaces are given.Second,the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in C~n are also established.In particular,the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings(include quasi-convex mappings of type A and quasi-convex mappings of type B) in several complex variables are get accordingly.Our results state that a weak version of the Bieberbach conjecture for quasi-convex mappings of type B and order α in several complex variables is proved,and the derived conclusions are the generalization of the classical results in one complex variable.     

An oscillating multiplier on Herz type spaces

In this paper, the boundedness of an oscillating multiplier m γ,β for different β on the Herz type spaces is obtained. This operator was initially studied by Wainger and Fefferman-Stein. Our results extend one of the main results in a paper by Xiaochun Li and Shanzhen Lu for the non-weighted case, if β is close to 1 or α is suitably large. For β ≥ 1, the results with no weights on the Herz type spaces are also new.     

New Subclasses of Biholomorphic Mappings and the Modified Roper-Suffridge Operator

The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The RoperSuffridge operator on the unit ball B~n in C~n is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator [Φ_(G,γ)(f)](z) preserves the properties of S_Ω~*(A, B), as well as strong and almost spirallikeness of type β and order α on B~n. Thus, the mappings in S_Ω~*(A, B), as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.     

A Remark on the Generalized Roper-Suffridge Extension Operator for Spirallike Mappings of Type β and Order α

Suppose f is a spirallike function of type β and order α on the unit disk D.let Ωn,p1,p2,…,pn={z=(z1,z2,…,zn)'∈Cn:nΣj=1|zj|pj<1},where 1≤p1≤2,pj≥1,j=2,…,n,are real numbers. In this paper,we will prove that Фn,β2,у2…βn,уn(f)(z)=(f(z1),(f(z1)z1)β2(f'(z1))у2z2,…,(f(z1)z1)βn(f'(z1))уnzn)' Preserves spirallikeness of type β and order α on Ωn,p1,p2,…pn.