THE NONLINEAR BOUNDARY VALUE PROBLEM FOR k HOLOMORPHIC FUNCTIONS IN C^(2)

k holomorphic functions are a type of generation of holomorphic functions.In this paper,a nonlinear boundary value problem for k holomorphic functions is primarily discussed on generalized polycylinders in C^(2).The existence of the solution for the problem is studied in detail with the help of the boundary properties of Cauchy type singular integral operators with a k holomorphic kernel.Furthermore,the integral representation for the solution is obtained.

NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS SHARING A HOLOMORPHIC FUNCTION AND THE CONVERSE OF THE BLOCH PRINCIPLE

In this paper, we investigate normal families of meromorphic functions, prove some theorems of normal families sharing a holomorphic function, and give a counterex- ample to the converse of the Bloch principle based on the theorems.

NORMAL FAMILIES AND SHARED VALUES OF HOLOMORPHIC FUNCTIONS

Let F be a family of holomorphic functions in a domain D, k be a positive integer, a, b（≠0）, c（≠0） and d be finite complex numbers. If, for each f∈F, all zeros of f-d have multiplicity at least k, f^（k） = a whenever f=0, and f=c whenever f^（k） = b, then F is normal in D. This result extends the well-known normality criterion of Miranda and improves some results due to Chen-Fang, Pang and Xu. Some examples are provided to show that our result is sharp.

SCHWARZ-PICK ESTIMATES FOR BOUNDED HOLOMORPHIC FUNCTIONS ON CLASSICAL DOMAINS

In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.

SCHWARZ-PICK ESTIMATES FOR PARTIAL DERIVATIVES OF ARBITARY ORDER OF BOUNDED HOLOMORPHIC FUNCTIONS IN THE UNIT BALL OF C^n

This paper gives a Schwarz-Pick estimate for bounded holomorphic functions in the unit ball of C n .

SOME REMARKS ON HOLOMORPHIC FUNCTIONS AND TAYLOR SERIES IN C^n

Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.

WEIGHTED INTEGRALS OF HOLOMORPHIC FUNCTIONS

Given an admissible weight w and 0<p<∞, the estimate∫ D|f(z)| pw(z)dm(z)～|f(0)| p+∫ D|f′(z)| p ψ p(z)w(z)dm(z)is valid for all holomorphic functions f in the unit disc D. Here,ψ(r)=∫ 1 rw(t)dtw(r) is the distortion of w. As an application of the above estimate, it is proved that the Cesàro operator C[·] is bounded on the weighted Bergman spaces L p a,w (D).

Normal Families of Holomorphic Functions Concerning Shared a Polynomial

In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k(2)be a positive integer,K be a positive number andα(z)be a polynomial of degree p(p 1).For each f∈F and z∈D,if f and f sharedα(z)CM and|f(k)(z)|K whenever f(z)-α(z)=0 in D, then F is normal in D.

Normality Criteria for Families of Holomorphic Functions

Let f be a holomorphic function on a domain D (?) C, and let a be a finite complex number. We denote by Ef(α) = {z∈ D : f(z) = a, ignoring multiplicity} the set of all distinct α-points of f. Let F be a family of holomorphic functions on D. If there exist three finite values a, b(≠ 0, a) and c(≠0) such that for every f ∈ F, Ef(0) (?) Ef'(a) and Ef'(b)(?) Ef(c), then F is a normal family on D.

Normal Families of Holomorphic Functions Concerning Zero Numbers

In this paper, we study the normal families related with a Hayman conjecture of higher derivative concerning zero numbers, and get one normal criteria.Our result improve some earlier related result.

The Refined Schwarz-Pick Estimates for Positive Real Part Holomorphic Functions in Several Complex Variables

In this article,the refined Schwarz-Pick estimates for positive real part holomorphic functions p(x)=p(0)+Σ_(m=k)^(∞)D^(M)p(0)(x^(m))/m!:G→Care given,where k is a positive integer,and G is a balanced domain in complex Banach spaces.In particular,the results of first order Fréchet derivative for the above functions and higher order Frechet derivatives for positive real part holomorphic functions p(x)=p(0)+Σ_(s=1)^(∞)D^(sk)p(0)(x^(sk))/(sk)!:G→Care sharp for G=B,where B is the unit ball of complex Banach spaces or the unit ball of complex Hilbert spaces.Their results reduce to the classical result in one complex variable,and generalize some known results in several complex variables.

Schwarz-Pick Estimates for Bounded Holomorphic Functions in the Unit Ball of C^n

We give a Schwarz-Pick estimate for bounded holomorphic functions on unit ball in Cn, and generalize some early work of Schwarz-Pick estimates for bounded holomorphic functions on unit disk in C.

A Criterion of Normality Based on a Single Holomorphic Function

Let F be a family of functions holomorphic on a domain D C C. Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k- 1, such that h（z） has no common zeros with any f∈F. Assume also that the following two conditions hold for every f ∈F ：（a） f（z） = 0 == f （z） = h（z）; and （b） y（z） = h（z） == ｜f（k）（z）｜ ≤ c, where c is a constant.Then F is normal on D.

Decomposition of L^P( Da) Space and Boundary Value of Holomorphic Functions

This paper deals with two topics mentioned in the title. First, it is proved that function f in L^P（ Da） can be decomposed into a sum g ＋ h, where Da is an angular domain in the complex plane, g and h are the non-tangential limits of functions in H^P（Da） and H^P（aD^c） in the sense of LP（Da）, respectively. Second, the sufficient and necessary conditions between boundary values of holomorphic functions and distributions in n-dimensional complex space are obtained.

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D C C, all of whose zeros have multiplicity at least k, where k 〉 2 is an integer. And let h（z）≠ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈F： （a） f（z） = 0 [f^（k）（z）｜ 〈 ｜h（z）｜; （b） f^（k）（z）≠ h（z）. Then F is normal on

Bounded Projections and Duality on Spaces of Holomorphic Functions in the Unit Ball of (?)~n

In this paper three Banach spaces A(ф),A(ф)and A~1(ф)of functions holomor- phic in the unit ball B of ■~n are defined.We exhibit bounded projections from C(B)onto A(ф),from L~1(B)onto A~1(ф),and from L~∞(B)onto A(ф).Using these projections,we show that A(ф)~*≌A~1(ф)and A~1(ф)~*≌A(ф).

Dimension Estimate of Polynomial Growth Holomorphic Functions

On a complete noncompact Kähler manifold M^(n)(complex dimension)with non-negative Ricci curvature and Euclidean volume growth,we prove that polynomial growth holomorphic functions of degree d has an dimension upper bound cdn,where c depends only on n and the asymptotic volume ratio.Note that the power is sharp.

SOME PROPERTIES OF HOLOMORPHIC CLIFFORDIAN FUNCTIONS IN COMPLEX CLIFFORD ANALYSIS

In this article, we mainly develop the foundation of a new function theory of several complex variables with values in a complex Clifford algebra defined on some subdomains of C^n＋l, so-called complex holomorphic Cliffordian functions. We define the complex holomorphic Cliffordian functions, study polynomial and singular solutions of the equation D△^mf= 0, obtain the integral representation formula for the complex holo-morphic Cliffordian functions with values in a complex Clifford algebra defined on some submanifolds of C^n＋1, deduce the Taylor expansion and the Laurent expansion for them and prove an invariance under an action of Lie group for them.

RATIONAL FUNCTIONS OF BEST UNIFORM APPROXIMATION AND HOLOMORPHIC CONTINUATION OF FUNCTIONS

Let f be a function, continuous and real valued on the segment △,△ (-∞,∞) and {Rn} be the sequence of the rational functions of best uniform approximation to fon △ of order (n,n). In the present work, the convergence of {Rn} in the complex plane is considered for the special caseswhen the poles (or the zeros, respectively) of {Rn} accumulate in the terms of weak convergence of measures to acompact set of zera capacity.As a consequence, sufficient conditions for the holomorphic and the meromorphic continuability of fare given.

The Construction and Analysis of Linear Ring Spaces

This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi-linearity is discussed, in addition to the examination of properties and classifications of linear ring spaces. Particularly, the ring of holomorphic functions on a region of the complex plane is examined, and the manner in which it generates an iterated linear ring space under the complex derivative operator. This notion is then generalized to all rings with nth order linear and surjective operators. Basic operator theory regarding the classifications of linear ring maps is also covered.