On the Fekete and Szeg? Problem for Starlike Mappings of Order α

Let S_α~*be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szeg? inequality for the class S_α~*, and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in C~n.     

On quasi-convex mappings of orderαin the unit ball of a complex Banach space Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper, a class of biholomorphic mappings named quasi-convex mapping of order a in the unit ball of a complex Banach space is introduced. When the Banach space is confined to Cn, we obtain the relation between this class of mappings and the convex mappings. Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space X. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order a defined on the polydisc in Cn and on the unit ball in a complex Banach space, respectively.     

The Pointwise Multipliers From Space F（p, q, s） to Bloch Type Space in C^n

Let B be the unit ball in C^n. By H(B) we denote the class of all holomorphic functions on B and H^∞ denotes the class of all bounded holomorphic functions on B.     

The Coefficient Inequalities for a Class of Holomorphic Mappings in Several Complex Variables

The authors establish the coefficient inequalities for a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in ■~n,which are natural extensions to higher dimensions of some Fekete and Szeg? inequalities for subclasses of the normalized univalent functions in the unit disk.     

The Coefficient Inequalities for a Class of Holomorphic Mappings in Several Complex Variable

The authors establish the coefficient inequalities for a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in ■ⁿ,which are natural extensions to higher dimensions of some Fekete and Szeg? inequalities for subclasses of the normalized univalent functions in the unit disk.     

ON THE VALIRON’S THEOREM IN THE POLYDISK

In this paper, we discuss the Valiron's theorem in the unit polydisk DN. We prove that for a holomorphic map φ : DN→ DNsatisfying some regular conditions, there exists a holomorphic map θ : DN→ H and a constant α 0 such that θoφ =1/aθ.It is based on the extension of Julia-Wolff-Carath′eodory(JWC) theorem of D in the polydisk.     

Coefficient Estimates for Certain Subclasses of Bi-Univalent Ma-Minda Mocanu-Convex Functions

In this paper, we introduce and investigate a new subclass of the function class Σ of bi-univalent functions of the Mocanu-convex type defined in the open unit disk, satisfy Ma and Minda subordination conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients |a2| and |a3| for functions in the new subclass introduced here. Further Application of Hohlov operator to this class is obtained. Several(known or new) consequences of the results are also pointed out.     

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 ...     

Analytic Properties for Holomorphic Matrix-valued Maps in C~(2×2)

In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on C~(2×2) and Julia set of one dimensional complex dynamic system.     

Sharp distortion theorems for a subclass of close-to-convex mappings

We introduce the class of strongly close-to-convex mappings of order c~ in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space X or the unit polydisc in Cn. As an application, a sharp growth theorem for strongly close-to-convex mappings of order α is obtained.     

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 .     

Dual Toeplitz operators on the sphere

Dual Toeplitz operators on the Hardy space of the unit circle are anti-unitarily equivalent to Toeplitz operators. In higher dimensions, for instance on the unit sphere, dual Toeplitz operators might behave quite differently and, therefore, seem to be a worth studying new class of Toeplitz-type operators. The purpose of this paper is to introduce and start a systematic investigation of dual Toeplitz operators on the orthogonal complement of the Hardy space of the unit sphere in Cn . In particular, we establish a corresponding spectral inclusion theorem and a Brown-Halmos type theorem. On the other hand, we characterize commuting dual Toeplitz operators as well as normal and quasinormal ones.     

A Vanishing Theorem on a Class of Hartogs Domain

In this paper, we consider the d-boundedness of the Bergman metric and a vanishing theorem of L²-cohomology on a class of Hartogs domain, whose base domain is the production of two irreducible bounded symmetric domains of the first type, by using the Bergman kernel function,invariant function, holomorphic automorphism group and so on.     

ON THE ASYMPTOTIC BEHAVIOR OF HOLOMORPHIC ISOMETRIES OF THE POINCARE DISK INTO BOUNDED SYMMETRIC DOMAINS

In this article we bounded symmetric domains study holomorphic isometries of the Poincare disk into Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometrics up to normalizing constants with respect to the Bergman metric, showing in particular that the graph 170 of any germ of holomorphic isometry of the Poincar6 disk A into an irreducible bounded symmetric domain Ω belong to C^N in its Harish-Chandra realization must extend to an affinealgebraic subvariety V belong to C × C^N = C^N＋1, and that the irreducible component of V ∩ （△ × Ω） containing V0 is the graph of a proper holomorphic isometric embedding F ： A→ Ω. In this article we study holomorphie isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δ△. Starting with the structural equation for holomorphic isometrics arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphie isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ‖σ‖ must vanish at a general boundary point either to the order 1 or to the order 1/2, called a holomorphie isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.     

DEGENERATE ELLIPTIC SYSTEMS OF CLASS Ⅰ

In this paper, we considered the first class (class Ⅰ) of elliptic systems of partial differential equations with a special kind of lower-order terms which are one-order singular. For the classical well-posedness of Dirichlet problem and skew derivative problem in the unit circle, we gave a discussion in full detail, and obtained complete results.     

HANKEL OPERATORS ON HARDY SPACES AND SCHATTEN CLASSES

For a holomorphic function f on the unit ball B~N of C~N, it is proved that the reduced Hankel oporator R_f on Hardy space H~2(B~N) is of Schatten class S_p for p≥1 if and only if f is in a corresponding Sobolev space.     

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.     

A REMARK ON KOLMOGOROV'S COMPARISON THEOREM

In approximation theory the theorem of Kolmogorov concerning the comparison of derivatives of differentiable functions defined on the real line is well-known. It plays an important r le in establishing sharp inequalities between the norms of derivatives of a function. In this note we establish a comparison theorem of Kolmogorov type on a class of functions which are defined on the real line and can be continuated analytically in a stripped region containing the real llne. As a consequence we have derived an inequality of Landau-Kolmogorov type on this function class, and moreover, we have applied it to get the exact estimation for the Kolmogorov's N-widths of the analytic function class.     

THE HAUSDORFF MEASURE OF SIERPINSKI CARPETS BASING ON REGULAR PENTAGON

In this paper,we address the problem of exact computation of the Hausdorff measure of a class of Sierpinski carpets E the self-similar sets generating in a unit regular pentagon on the plane.Under some conditions,we show the natural covering is the best one,and the Hausdorff measures of those sets are euqal to | E | s,where s=dim H E.     

Bloch constant of holomorphic mappings on the unit polydisk of C~n

In this paper,we give a definition of Bloch mappings defined in the unit polydisk D~n, which generalizes the concept of Bloch functions defined in the unit disk D.It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables.We shall establish the corresponding distortion theorems for subfamiliesβ(K)andβ_(loc)(K) of Bloch mappings defined in the polydisk D~n,which extend the distortion theorems of Liu and Minda to higher dimensions.As an application,we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloeh mappings defined in D~n.In particular,our results reduce to the classical results of Ahlfors and Landau when n=1.