A NOTE ON SCHWARZ LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS ON D

In this note, we consider a holomorphic mapping f from the unit disk C in C to p-ball B^p = {z∈C^n;i=1∑n｜zi｜p〈1,1〈p〈＋∞. It is proved that for such f,｜ ｜ ｜f｜｜（z）｜≤1-｜｜f（z）｜｜^2/1-｜z｜^2,z∈D. The extremal problem is also discussed when p is an even number. This result extends some related results on Schwarz lemma.

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.

THE ∂∂^(-)-BOCHNER FORMULAS FOR HOLOMORPHIC MAPPINGS BETWEEN HERMITIAN MANIFOLDS AND THEIR APPLICATIONS

In this paper,we derive some∂∂^(-)-Bochner formulas for holomorphic maps between Hermitian manifolds.As applications,we prove some Schwarz lemma type estimates,and some rigidity and degeneracy theorems.For instance,we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive(resp.non-negative)ℓ-second Ricci curvature to a Hermitian manifold with non-positive(resp.negative)real bisectional curvature.These theorems generalize the results[5,6]proved recently by L.Ni on Kähler manifolds to Hermitian manifolds.We also derive an integral inequality for a holomorphic map between Hermitian manifolds.

Weighted Integral Means of Mixed Areas and Lengths Under Holomorphic Mappings

This note addresses monotonic growths and logarithmic convexities of the weighted （（1-t2）αdt2, -∞〈α〈∞, 0〈t〈1） integral means Aα,β（ f ,·） and Lα,β（ f ,·） of the mixed area （πr2）-βA（ f ,r） and the mixed length （2πr）-βL（ f ,r） （0≤β≤1 and 0〈r〈1） of f （rD） and？f （rD） under a holomorphic map f from the unit disk D into the finite complex plane C.

Properties for a Class of Holomorphic Mapson Hilbert Space

In this note we deal with a class of holomorphic maps with generalized positive real part on Hilbert space. The distortion theorem and Pick Julia type theorem for these maps are obtained.

On the Proper Holomorphic Mappings between Equidimensional Hartogs Domains over Hermitian Symmetric Manifolds

In this paper,we study a family of Hartogs domains fibred over Hermitian symmetric manifolds being a unit ball in C^(m).The aim of the present study is to establish the rigidity results about proper holomorphic mappings between two equidimensional Hartogs domains over Hermitian symmetric manifolds.In particular,we can fully determine its biholomorphic equivalence and automorphism group.

ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO P^n(C)

This article proves the existence of Julia directions of value distribution of holomorphic mapping f from the unit disk into the n-dimensional complex projective spacePn（C） under the assumption limsupT（r,f）/log 1/1-r = ＋∞ for hypersurfaces in general position. A heuristic principle concerning the existence of Julia directions of holomorphic mappings from the unit disk into Pn（C） is given also.

NORMAL CRITERION FOR FAMILIES OF HOLOMORPHIC MAPS OF SEVERAL COMPLEX VARIABLES INTO P^N(C)WITH MOVING HYPERSURFACES

This article gives a normal criterion for families of holomorphic mappings of several complex variables into P N(C)for moving hypersurfaces in pointwise general position,related to an Eremenko’s theorem.

Extension of Proper _-holom orphic Mappings

In this paper,we prove that a proper μ holomorphic mapping f:D 1→D 2 between bounded domains with some convexity,such that f satisfies some growth condition,extends smoothly to bD 1-{z:U(z)=0}.

SOME FURTHER RESULTS FOR HOLOMORPHIC MAPS ON PARABOLIC RIEMANN SURFACES

In [1],they generalized R.Nevanlinna’s results to Y,where Y is a parabolic Riemann Surface.In this paper,following their method,we develop some further results for holomorphic maps on Y,including the maps into Pn(C),the complex projective varieties,and Abelian varieties.

Some Growth Rates of Certain Holomorphic Maps

In this paper we discuss some properties of h olomorphic maps between Kahler manifolds with certain curvature restrictions.

Classification of Proper Holomorphic Mappings between Hartogs Domains over Homogeneous Siegel Domains

The Hartogs domain over homogeneous Siegel domain D_(N,s)(s>0)is defined by the inequality■,where D is a homogeneous Siegel domain of typeⅡ,(z,ζ)∈D×C~N and KD(z,z)is the Bergman kernel of D.Recently,Seo obtained the rigidity result that proper holomorphic mappings between two equidimensional domains D_(N,s)and D'_(N',s')are biholomorphisms for N≥2.In this article,we find a counter-example to show that the rigidity result is not true for D_(1,s)and obtain a classification of proper holomorphic mappings between D_(1,s)and D'_(1,s').

ON THE GENERALIZED FRITZ JOHN OPTIMALITY CONDITIONS OF VECTOR OPTIMIZATION WITH SET-VALUED MAPS UNDER BENSON PROPER EFFICIENCY

A kind of tangent derivative and the concepts of strong and weak * pseudoconvexity for a set-valued map are introduced. By the standard separation theorems of the convex sets and cones the optimality Fritz John condition of set-valued optimization under Benson proper efficiency is established, its sufficience is discussed. The form of the optimality conditions obtained here completely tally with the classical results when the set-valued map is specialized to be a single-valued map.

ON THE SINGULAR DIRECTIONS OF VALUE DISTRIBUTION OF HOLOMORPHIC CURVES IN P^n(C)

This article proves the existence of singular directions of value distribution theory for some transcendental holomorphic curves in the n-dimensional complex projective space P^n（C）.. An example is given to complement these results.

Benson proper efficiency for vector optimization of generalized subconvexlike set-valued maps in ordered linear spaces

Several equivalent statements of generalized subconvexlike set-valued map are established in ordered linear spaces. Using vector closure, we introduce Benson proper efficient solution of vector optimization problem. Under the assumption of generalized subconvexlikeness, scalarization, multiplier and saddle point theorems are obtained in the sense of Benson proper efficiency.

DISTORTION THEOREMS FOR BLOCH MAPPINGS ON THE UNIT POLYDISC D^n

In this article, we establish distortion theorems for some various subfamilies of Bloch mappings defined in the unit polydisc Dn with critical points, which extend the results of Liu and Minda to higher dimensions. We obtain lower bounds of ｜ det（f＇（z））｜and Rdet（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.

Holomorphic Curves into ■~N(C) That Share a Set of Moving Hypersurfaces

Let F be a family of holomorphic curves of a domain D in C into a closed subset X in ■~N(C). Let Q_1(z),…, Q_(2t+1)(z) be moving hypersurfaces in ■~N(C) located in pointwise t-subgeneral position with respect to X. If each pair of curves f and g in F share the set {Q_1(z),…, Q_(2t+1)(z)}, then F is normal on D. This result greatly extend some earlier theorems related to Montel's criterion.

Co-commuting Mappings of Generalized Matrix Algebras

Let G be a generalized matrix algebra over a commutative ring R and Z（G） be the center of G. Suppose that F, T ：G→G are two co-commuting R-linear mappings, i.e., F（x）x = xT（x） for all x ∈ G. In this note, we study the question of when co-commuting mappings on G are proper.

Results About Parabolic-Like Mappings

In this paper we present the most important definitions and results of the theory of parabolic-like mappings, and we will give an example. The proofs of the results can be found in [2,4] and [3].

NORMAL CRITERIA FOR A FAMILY OF HOLOMORPHIC CURVES

In this paper,we extend the concept of holomorphic curves sharing hyperplanes and introduce definitions of restricted hyperplanes and partial shared hypersurfaces.Then,we prove several normal criteria of the family of holomorphic curves and holomorphic mappings that concern restricted hyperplanes and partial shared hypersurfaces.These results generalize the Montel-type normal criterion of holomorphic curves.