Distortion theorems for subclasses of starlike mappings along a unit direction in ℂ

In this paper, we obtain a distortion theorem of Jacobian matrix for biholomorphic subclasses of starlike mappings along a unit direction on the unit polydisc. These results extend the classical distortion theorem of starlike mappings to higher dimensions. We then give an upper bound estimate for biholomorphic subclasses of starlike mappings along a unit direction in a complex Banach space.     

UNIFORMLY STARLIKE MAPPINGS AND UNIFORMLY CONVEX MAPPINGS ON THE UNIT BALL B^n

Abstract In this article, we extend the definition of uniformly starlike functions and uni- formly convex functions on the unit disk to the unit ball in C^n, give the discriminant criterions for them, and get some inequalities for them.     

MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS

In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n：｜z1｜^2＋n∑j=2｜zj｜^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈（-π/2,π/2）.Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F（z）=（f（z1）＋f＇（z1）Pm（z0）,（f＇（z1））^1/mz0）＇where f is a normalized biholomorphic function on the unit disc D, z = （z1,z0） ∈Ωn1p2,…,pn,z0=（z2,…,zn）∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.     

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

In this paper,we define the class■ of g-parametric star like mappings of real order γ on the unit ball B_X in a complex Banach space X,where g is analytic and satisfies certain conditions.By establishing the distortion theorem of the Fréchet-derivative type of■ with a weak restrictive condition,we further obtain the distortion results of the Jacobi-determinant type and the Fréchet-derivative type for the corresponding classes(compared with■) defined on the unit polydisc(resp.unit ball ...     

THE GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS

1.IntroductionOnthegeometricfunctiontheoryofonecomplexvariable,thefollowinggrowthandi-coveringtheoremiswellknown(see[2]).TheoremA.Foreachno~alizedunivalentjunctionfontheunitdiscDCC,ESPecially,theleft-handsideOftheaboveinequalityimpliesf(D)2ID.ForeachzED,z/0,equalityoccursintheaboveinequalityifandonlyiffisKoe6efunctionK(z)=theoritsrotatione--"K(e"z).Itisnaturaltoextendthisandotherresultsonthegeometricfunctiontheoryofonevariablestoseveralvariables.Butasearlyasfiftyyearsago,H.Cartanpointe…     

DISTORTION THEOREMS FOR BLOCH MAPPINGS ON THE UNIT POLYDISC Dn

In this article,we establish distortion theorems for some various subfamilies of Bloch mappings defined in the unit polydisc D n with critical points,which extend the results of Liu and Minda to higher dimensions.We obtain lower bounds of | det(f′(z))| and det(f′(z)) for Bloch mapping f.As an application,some lower and upper bounds of Bloch constants for the subfamilies of holomorphic mappings are given.     

COINCIDENCE THEOREMS INVOKING COMPOSITES OF ACYCLIC MAPPINGS AND APPLICATIONS

1IntroductionRecentlytherehaveappearedanumberofnewfixedpointtheoremsandcoincidencetheorexnsinvolvingsthcalledKakutanifactorizablemultifunctionsorcompositesofacyclicset-valuedmappingsdefinedonconvexsubsetsoftopologicalvectorspaces;forexample,see[1-3].Intheseresults,authorsassumethateitherthedoesnOfmappingiscompact,orthemappillgiscompact.Inthispaper,wefirstprovesomeverygeneralcoincidencetheoremsinvolvingtheclassVJofset-valuedmappings"lidernoncompactsettingwhichgeneralizesTheoremIofPark,Singha…     

DISTORTION THEOREMS FOR BIHOLOMORPHIC CONVEX MAPPINGS ON BOUNDED CONVEX CIRCULAR DOMAINS

1.TwoDifferentDistortionTheoremsVariousdistortiontheoremsforfamiliesofunivalentfunctionshavebeenstudiedsinceasearlyas1907whenK5bediscoveredhis"Verzerrungssatz",thedistortiontheoremfortheclassofunivalentfunctionsdefinedontheunitdiskillthecomplexplaneC.LetfiCC"beadomainandf(z)=(fi(z),.',m(z))beabiholomorphicmappingonfiwhichmapsfitoC".Therearemanycounter-examplestoshowthatIdetJj(z)landJI(z)Jj(z)'havenofiniteupperboundandnonon--zerolowerbound,whereJf(z)istheJacobianoffatpointz.LetfiCC"bea…     

SHARP DISTORTION THEOREMS FOR A CLASS OF BIHOLOMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES

In this paper,we first establish the sharp growth theorem and the distortion theorem of the Frechet derivative for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in Cⁿ with some restricted conditions.We next give the distortion theorem of the Jacobi determinant for biholomorphic mappings defined on the unit ball of Cⁿ with an arbitrary norm and the unit polydisk in Cⁿ under certain restricted assumptions.Finally we ...     

COINCIDENCE THEOREMS INVOLVING COMPOSITES OF ACYCLIC MAPPINGS AND APPLICATIONS

Some new coincidence theorems involving a newclass of set-valued mappings containing composites of acyclic mappings definedon a contractible space are proved. As applications, some existence theoremsof maximal elements and solutions of abstract variational inequalities, andbest approximation theorems are proved. These theorems improve and generalizea number of known results in recent literature.     

模糊度量空间中多变量压缩映射的公共耦合不动点定理

本文提出了多变量映射的一致点的概念.本文的结果是模糊偏序度量空间中不动点定理的一些主要结论从低维到高维的推广.     

DISTORTION THEOREM FOR BIHOLOMORPHIC MAPPINGS IN TRANSITIVE DOMAINS （Ⅳ）

The distortion theorem for biholomorphic starlike mappings (with respect to origin )in bounded symmetric domains are given,The distortion theorem for locally biholomorphic convex mappings in bounded symmetric domains are given also.     

APPROXIMATION OF FIXED POINTS AND VARIATIONAL SOLUTIONS FOR PSEUDO-CONTRACTIVE MAPPINGS IN BANACH SPACES

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gteaux differentiable norm. Assume that every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive mappings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, convex optimization problems, and split feasibility problems. Our result extends many recent important results.     

A SCHWARZ-PICK LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS FROM THE POLYDISK INTO THE UNIT BALL

In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings from the polydisk into the unit ball. This result extends some related results.     

多复变数完全拟凸映射的分解定理

设Ω_1C~(n1),Ω_2C~(n2)为凸的Reinhardt域,f(z,w)=(f1(z,w),f2(z,w))'为Ω_1×Ω_2上的正规化全纯映射.本文证明f为Ω_1×Ω_2上的正规化双全纯完全拟凸映射当且仅当 f(z,w)=(Φ_1(z),Φ_2(w))'其中φj:Ωj→C~(nj)是Ωj(j=1,2)上的正规化双全纯完全拟凸映射。     

The Study for Some Subclasses of Biholomorphic Mappings by an Unified Method

LetΩ∈Cn be a bounded starlike circular domain with 0∈Ω. In this paper, we introduce a class of holomorphic mappings Mg onΩ. Let f(z) be a normalized locally biholomorphic mapping onΩsuch that Jf-1(z)f(z)∈Mg and z=0 is the zero of order k + 1 of f(z)-z. We obtain the growth and covering theorems for f(z). Especially, as corollaries, we unify and generalize many known results. Moreover, in view of proofs of corollaries, the essential relations among the subclasses of starlike mappings are shown.     

THE PARAMETRIC REPRESENTATION FOR SPIRAL-LIKE MAPPINGS OF TYPE α ON BOUNDED BALANCED PSEUDOCONVEX DOMAINS

Let Ω be a bounded balanced pseudoconvex domain in Cn that admits a plurisubhamonic defining function of class C2. The authors prove that if f is a spiral-like mapping of type α on Ω, then f can be expressed as a limiting form concerning a Schwarz mapping. Moreover, the characteristics for star-like mappings and spiral-like mappings of type α are obtained. As a direct application, the growth theorem for spiral-like mappings of type α is set up.     

关于强星像函数类的一个子类

引入了强星像函数类的一个子类Ｓ（α，β，ρ，σ，Ａ，Ｂ），讨论了它的从属关系、包含关系、星像阶数和强星像阶数以及积分表示定理等，所得结果推广了一些作者的相关工作。     

CARLESON MEASURES FOR BESOV-SOBOLEV SPACES WITH APPLICATIONS IN THE UNIT BALL OF C^n

This paper is devoted to give the connections between Carleson measures for Besov-Sobolev spaces Bσp (B) and p-Carleson measure in the unit ball of Cn. As appli-cations, we characterize the Riemann-Sti...