Several Complex Variables Research Articles

Inequality for entire functions involving their maximum modulus and maximum term

An estimate of the Wiman-Valiron type for a maximum modulus on a polydisk of an entire function of several complex variables is obtained. The estimate contains a weight function involved also in the calculation of the radius of the admissible ball.

DISTORTION THEOREMS OF STARLIKE MAPPINGS IN SEVERAL COMPLEX VARIABLES

In this paper, we first obtain a distortion theorem of Jacobian matrix $J_f(z)$ for biholomorphic starlike mapping $f$ along a unit direction on the unit polydisc. This result extends the classical distortion theorem of starlike functions to higher dimensions. We then give an upper bound estimate of distortion theorem for biholomorphic starlike mappings along a unit direction in a complex Banach space. Finally, we propose two conjectures on starlike mappings in several complex variables.

Growth and distortion theorems on subclasses of quasi-convex mappings in several complex variables

In this paper, the refining growth and covering theorems for f are established, where f is a quasi-convex mapping of order α and x = 0 is a zero of order k + 1 of f(x) − x. As an application, we obtain the upper and lower bounds on the distortion theorem of f(x) defined on the unit polydisc of ℂn. The upper bound of the distortion theorem for f(x) defined on the unit ball of a complex Banach space is also given. Our results extend the growth and distortion theorems for convex functions of one complex variable to quasi-convex mappings of several complex variables.

Growth and approximation of generalized biaxially symmetric potentials in several complex variables

The paper deals with the aspect of the growth of entire functions of several complex variables. The growth of the entire functions with respect to each of the variables separately has been studied by defining partial order and partial type. Finally, we have studied the growth and polynomial approximation of entire function generalized biaxially symmetric potential with respect to each of the variables separately.

Unicity of meromorphic solutions of partial differential equations

In this survey, results on the existence, growth, uniqueness, and value distribution of meromorphic (or entire) solutions of linear partial differential equations of the second order with polynomial coefficients that are similar or different from that of meromorphic solutions of linear ordinary differential equations have been obtained. We have characterized those entire solutions of a special partial differential equation that relate to Jacobian polynomials. We prove a uniqueness theorem of meromorphic functions of several complex variables sharing three values taking into account multiplicity such that one of the meromorphic functions satisfies a nonlinear partial differential equations of the first order with meromorphic coefficients, which extends the Brosch’s uniqueness theorem related to meromorphic solutions of nonlinear ordinary differential equations of the first order.

DISTORTION THEOREMS OF STARLIKE MAPPINGS IN SEVERAL COMPLEX VARIABLES

In this paper, we first obtain a distortion theorem of Jacobian matrix $J_f(z)$ for biholomorphic starlike mapping $f$ along a unit direction on the unit polydisc. This result extends the classical distortion theorem of starlike functions to higher dimensions. We then give an upper bound estimate of distortion theorem for biholomorphic starlike mappings along a unit direction in a complex Banach space. Finally, we propose two conjectures on starlike mappings in several complex variables.

On Approximation and Generalized Type of Analytic Functions of Several Complex Variables

In the present paper, we study the polynomial approximation of analytic functions of several complex variables. The characterizations of generalized type of analytic functions of several complex variables have been obtained in terms of approximation and interpolation errors.

Algebraic dependences of meromorphic mappings in several complex variables

We give some theorems on algebraic dependence of meromorphic mappings in several complex variables into complex projective spaces.

Proper analytic free maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations – they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain D ˜ in g ˜ variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of D ˜ . Assuming that both domains contain 0, we show that if f : D → D ˜ is a proper analytic free map, and f ( 0 ) = 0 , then f is one-to-one. Moreover, if also g = g ˜ , then f is invertible and f − 1 is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and D ˜ .

Factorisations of the Helmholtz Operator, Radó’s Theorem, and Clifford Analysis

Rado’s theorem for holomorphic functions asserts that if a continuous function is holomorphic on the complement of its zero locus, then it is holomorphic everywhere. We prove in this paper an equivalent theorem for functions lying in the kernel of a first order differential operator \({\mathcal{D}}\) such that the Helmholtz operator ∇2+λ can be factorized as the composition \({\widehat{\mathcal{D}}\mathcal{D}}\) . We also analyse the factorisations \({\widehat{\mathcal{D}}\mathcal{D}}\) of the Laplace and Helmholtz operators associated to the Clifford analysis and the representations of holomorphic function of several complex variables.

UNICITY THEOREMS WITH TRUNCATED MULTIPLICITIES OF MEROMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES SHARING SMALL IDENTICAL SETS FOR MOVING TARGETS

In this article, unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables sharing small identical sets for moving targets are given.

Linear Differential Elimination for Analytic Functions

This paper provides methods to decide whether a given analytic function of several complex variables is a linear combination of finitely many given analytic functions with coefficients of the following special form: Each one of these coefficients is a composition of an unknown analytic function of less arguments than the function to be expressed, with fixed analytic functions. Methods which compute suitable coefficient functions in the affirmative case are presented as well.

Free Biholomorphic Classification of Noncommutative Domains

In this paper, we develop a theory of free holomorphic functions on noncommutative Reinhardt domains , generated by positive regular free holomorphic functions f in n noncommuting variables and by positive integers m ≥ 1, where is the algebra of all bounded linear operators on a Hilbert space . Noncommutative Berezin transforms are used to study Hardy algebras H ∞ (D f, rad m ) and domain algebras A(D f, rad m ) associated with and compositions of free holomorphic functions. We obtain noncommutative Cartan type results for formal power series, in several noncommuting indeterminates, which leave invariant the nilpotent parts of the corresponding domains. As a consequence, we characterize the set of all free biholomorphic functions with F(0) = 0. We show that the free biholomorphic classification of the domains is the same as the classification, up to unital completely isometric isomorphisms having completely contractive hereditary extension, of the corresponding noncommutative domain algebras A(D f, rad m ). In particular, we prove that Ψ : A(D f, rad 1 ) → A(D g, rad 1 ) is a unital completely isometric isomorphism if and only if there is a free biholomorphic map φ ∈ Bih(D g 1 , D f 1 ) such that This implies that the noncommutative domains and are free biholomorphic equivalent if and only if the domain algebras A(D f, rad 1 ) and A(D g, rad 1 ) are completely isometrically isomorphic. Using the interaction between the theory of functions in several complex variables and our noncommutative theory, we provide several results concerning the free biholomorphic classification of the noncommutative domains and the classification, up to completely isometric isomorphisms, of the associated noncommutative domain (resp. Hardy) algebras. In particular, we characterize the unit ball of among the noncommutative domains , up to free biholomorphisms. We also obtain characterizations for the unitarily implemented isomorphisms of noncommutative Hardy (resp. domain) algebras in terms of free biholomorphic functions between the corresponding noncommutative domains.

On interpolation for Hörmander’s algebras

We discuss some recent results on interpolation problems for weighted Hormander’s algebras of holomorphic functions in several complex variables, and also give a sharp estimate on counting functions of interpolating varieties.

Montel-type theorems in several complex variables with continuously moving targets

The authors introduce a new idea related to Montel-type theorems in higher dimension and prove some Montel-type criteria for normal families of holomorphic mappings and normal holomorphic mappings of several complex variables into P N (ℂ) for continuously moving hyperplanes in pointwise general position. The main results are also true for continuously moving hypersurfaces in pointwise general position. Examples are given to show the sharpness of the results.

On some classes of bounded univalent mappings in several complex variables

Let Bn be the Euclidean unit ball in Cn. In this paper, we study several properties of strongly starlike mappings of order α (0 < α < 1) and bounded convex mappings on Bn. We prove that K-quasiregular strongly starlike mappings of order α on Bn have continuous and univalent extensions to \({\overline{B}^n}\). We show that bounded convex mappings on Bn are strongly starlike of some order α. We give a coefficient estimate for K-quasiregular strongly starlike mappings of order α on Bn. Finally, we give examples of strongly starlike mappings of order α and bounded convex mappings on Bn.

Generalized Order and Best Approximation of Entire Function in Lp‐Norm

The aim of this paper is the characterization of the generalized growth of entire functions of several complex variables by means of the best polynomial approximation and interpolation on a compact K with respect to the set Ωr = {z ∈ Cn; exp VK(z) ≤ r}, where is the Siciak extremal function of a L‐regular compact K.

On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis

Euclidean Clifford analysis is a higher dimensional function theory, refining harmonic analysis, centred around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator, called the Dirac operator. More recently, Hermitean Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions of two Hermitean Dirac operators, invariant under the action of the unitary group. In this paper, a Cauchy integral formula is established by means of a matrix approach, allowing the recovering of the traditional Martinelli-Bochner formula for holomorphic functions of several complex variables as a special case.

GENERALIZATION OF THE FROBENIUS THEOREM ON INVOLUTIVITY

Given a system of s independent 1-forms on a smooth manifold M of dimension m, we study the existence of integral manifolds by means of various generalized versions of the Frobenius theorem. In particular, we present necessary and sufficient conditions for there to exist s'-parameter (s' < s) family of integral manifolds of dimension p := m-s, and a necessary and sufficient condition for there to exist integral manifolds of dimension p', p' <TEX>$\leq$</TEX> p. We also present examples and applications to complex analysis in several variables.

From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept

Historical work on the emergence of sheaf theory has mainly concentrated on the topological origins of sheaf cohomology in the period from 1945 to 1950 and on subsequent developments. However, a shift of emphasis both in time-scale and disciplinary context can help gain new insight into the emergence of the sheaf concept. This paper concentrates on Henri Cartan’s work in the theory of analytic functions of several complex variables and the strikingly different roles it played at two stages of the emergence of sheaf theory: the definition of a new structure and formulation of a new research programme in 1940–1944; the unexpected integration into sheaf cohomology in 1951–1952. In order to bring this two-stage structural transition into perspective, we will concentrate more specifically on a family of problems, the so-called Cousin problems, from Poincare (1883) to Cartan. This medium-term narrative provides insight into two more general issues in the history of contemporary mathematics. First, we will focus on the use of problems in theory-making. Second, the history of the design of structures in geometrically flavoured contexts—such as for the sheaf and fibre-bundle structures—which will help provide a more comprehensive view of the structuralist moment, a moment whose algebraic component has so far been the main focus for historical work.