Several Complex Variables Research Articles

A Refinement of the Coefficient Inequalities for a Subclass of Starlike Mappings in Several Complex Variables

In this paper, we establish the Fekete and Szego inequality for a normalized starlike function f on $${\mathbb {U}}$$ such that $$z=0$$ is a zero of order $$k+1$$ of $$f(z)-z$$, and then we extend this result to higher dimensions.

Unicity Theorems with Truncated Multiplicities of Meromorphic Mappings in Several Complex Variables for Few Fixed Targets

The purpose of our paper is twofold. Our first aim is to prove a uniqueness theorem for meromorphic mappings of ℂn into ℙN(ℂ) sharing 2N + 2 hyperplanes in the general position with truncated multiplicities, where all common zeros with multiplicities greater than a certain number do not need to be counted. Second, we consider the case of mappings sharing less than 2N +2 hyperplanes. These results improve some recent results.

Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions

Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density.The techniques combine methods from several complex variables (estimates for ∂¯) and the theory of localized frames in general reproducing kernel Hilbert spaces (with no analyticity assumed). The abstract results on Fekete points and deformation of frames may be of independent interest.

Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .

Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables

Abstract Let S ψ ∗ $\begin{array}{} \mathcal {S}^*_\psi \end{array}$ be a subclass of starlike functions in the unit disk 𝕌, where ψ is a convex function such that ψ(0) = 1, ψ′(0) > 0, ℜ(ψ(ξ)) > 0 and ψ(𝕌) is symmetric with respect to the real axis. We obtain the sharp solution of Fekete-Szegö problem for the family S ψ ∗ $\begin{array}{} \mathcal {S}^*_\psi \end{array}$ , and then extend the result to the case of corresponding subclass defined on the bounded starlike circular domain Ω in several complex variables, which give an unified answer of Fekete-Szegö problem for the kinds of subclasses of starlike mappings defined on Ω. At last, we propose two conjectures related the same problems on the unit ball in a complex Banach space and on the unit polydisk in ℂ n .

Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables

We investigate the slice holomorphic functions of several complex variables that have a bounded \(L\)-index in some direction and are entire on every slice \(\{z^0+t\mathbf{b}: t\in\mathbb{C}\}\) for every \(z^0\in\mathbb{C}^n\) and for a given direction \(\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}\). For this class of functions, we prove some criteria of boundedness of the \(L\)-index in direction describing a local behavior of the maximum and minimum moduli of a slice holomorphic function and give estimates of the logarithmic derivative and the distribution of zeros. Moreover, we obtain analogs of the known Hayman theorem and logarithmic criteria. They are applicable to the analytic theory of differential equations. We also study the value distribution and prove the existence theorem for those functions. It is shown that the bounded multiplicity of zeros for a slice holomorphic function \(F:\mathbb{C}^n\to\mathbb{C}\) is the necessary and sufficient condition for the existence of a positive continuous function \(L: \mathbb{C}^n\to\mathbb{R}_+\) such that \(F\) has a bounded \(L\)-index in direction.

Degeneracy and Finiteness Theorems for Meromorphic Mappings in Several Complex Variables

In this article, we prove that there are at most two meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 2)$ sharing $2n+2$ hyperplanes in general position regardless of multiplicity, where all zeros with multiplicities more than certain values do not need to be counted. We also show that if three meromorphic mappings $f^1,f^2,f^3$ of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 5)$ share $2n+1$ hyperplanes in general position with truncated multiplicity then the map $f^1\times f^2\times f^3$ is linearly degenerate.

Sum and product theorems relating to relative (p, q)-th Gol’dberg order, relative (p, q)-th Gol’dberg type and relative (p, q)-th Gol’dberg weak type of entire functions of several variables

In this paper we wish to establish some basic properties of entire functions of several complex variables on the basis of relative (p, q)-th Gol’dberg order, relative (p, q)-th Gol’dberg type and relative (p, q)-th Gol’dberg weak type where p and q are any two positive integers.

Subclasses of Biholomorphic Mappings Under the Extension Operators

In this article, we mainly study the invariance of some biholomorphic mappings with special geometric characteristics under the extension operators. First we generalize the Roper-Suffridge extension operators on Bergman-Hartogs domains. Then, by the geometric characteristics of subclasses of biholomorphic mappings, we conclude that the modified Roper-Suffridge operators preserve the properties of SΩ* (β,A,B), parabolic and spirallike mappings of type β and order ρ, strong and almost spirallike mappings of type β and order α as well as almost starlike mappings of complex order λ on $$\Omega_{p_{1}}^{B^{n}},\ldots,_{p_{s},q}$$ under different conditions, respectively. The conclusions provide new approaches to construct these biholomorphic mappings in several complex variables.

Monotone FISTA with Variable Acceleration for Compressed Sensing Magnetic Resonance Imaging.

An improvement of the monotone fast iterative shrinkage-thresholding algorithm (MFISTA) for faster convergence is proposed. Our motivation is to reduce the reconstruction time of compressed sensing problems in magnetic resonance imaging. The proposed modification introduces an extra term, which is a multiple of the proximal-gradient step, into the so-called momentum formula used for the computation of the next iterate in MFISTA. In addition, the modified algorithm selects the next iterate as a possibly-improved point obtained by any other procedure, such as an arbitrary shift, a line search, or other methods. As an example, an arbitrary-length shift in the direction from the previous iterate to the output of the proximal-gradient step is considered. The resulting algorithm accelerates MFISTA in a manner that varies with the iterative steps. Convergence analysis shows that the proposed modification provides improved theoretical convergence bounds, and that it has more flexibility in its parameters than the original MFISTA. Since such problems need to be studied in the context of functions of several complex variables, a careful extension of FISTA-like methods to complex variables is provided.

Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non-commuting variables. In this paper we introduce the class of \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-dimensional nc-universe $\m^d$}. We illustrate the use of such manifolds in free analysis through the construction of the non-commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non-commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in $\m^2$ we construct a 2-dimensional non-commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton-Girard formulae for power sums of two non-commuting variables.

Attractors of Compactly Generated Semigroups of Regular Polynomial Mappings

We investigate the metric space of pluriregular sets as well as the contractions on that space induced by infinite compact families of proper polynomial mappings of several complex variables. The topological semigroups generated by such families, with composition as the semigroup operation, lead to the constructions of a variety of Julia-type pluriregular sets. The generating families can also be viewed as infinite iterated function systems with compact attractors. We show that such attractors can be approximated both deterministically and probabilistically in a manner of the classic chaos game.

An inversion formula for the transport equation in ℝ3 using complex analysis in several variables

Abstract In this paper, the stationary photon transport equation has been extended by analytical continuation from ℝ 3 {\mathbb{R}^{3}} to ℂ 3 {\mathbb{C}^{3}} . A solution to the inverse problem posed by this equation is obtained on a hyper-sphere and a hyper-cylinder as X-ray and Radon transforms, respectively. We show that these results can be transformed into each other, and they agree with known results. Numerical reconstructions of a three-dimensional Shepp–Logan head phantom using the obtained inverse formula illustrate the analytical results obtained in this manuscript.

Generalized order and related growth measure of composite entire function of several complex variables on the basis of central index

Generalized order and related growth measure of composite entire function of several complex variables on the basis of central index

A New Cauchy Integral Formula in the Complex Clifford Analysis

In this paper, we construct an analogue of Bochner–Martinelli kernel based on theory of functions of several complex variables in complex Clifford analysis, which has generalized complex differential forms with Clifford basis vectors. Using these complex differential forms, we obtain the Stoke’s formula of complex Clifford functions which are defined on a domain \(\Omega \subset C^{n+1}\) and take values in a complex Clifford algebra \(Cl_{0,n}(C)\). Then, we give a Stoke’s formula which has a classical form and an analogue of Cauchy–Pompeiu formula which is represented by Bochner–Martinelli kernel, and establish an analogue of Cauchy integral formula in complex Clifford analysis. It is possible to promote these results to complex manifold’s corresponding results in the Clifford analysis using the representation by generalized complex differential forms.

The Estimates of All Homogeneous Expansions for a Subclass of ε Quasi-convex Mappings in Several Complex Variables

The authors obtain the estimates of all homogeneous expansions for a subclass of e quasi-convex mappings on the unit ball in complex Banach spaces. Moreover, the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in ℂn are also obtained. Especially, the above estimates are only sharp for a subclass of starlike mappings, quasi-convex mappings and quasi-convex mappings of type $$\mathbb{A}$$ . The results are the generalization of many known results.

Nonlocal Boundary-Value Problem for a Differential-Operator Equation with Weak Nonlinearity in the Spaces of Dirichlet–Taylor Series with Fixed Spectrum

We study the nonlocal boundary-value problem for a differential-operator equation with nonlinear right-hand side and an operator B = (B1,…B p ), where its components B j ≡ z j ∂/∂z j , j = 1,…,p, are the operators of generalized differentiation with respect to the complex variable z j . By using the Nash–Moser iterative scheme, we establish the conditions of solvability of this problem in the scale of spaces of functions of several complex variables, which are Dirichlet–Taylor series with fixed spectrum.

Bavrin\u2019s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of mathbb {C}^{n}

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families mathcal {C}_{mathcal {G}},mathcal {M}_{mathcal {G}},mathcal {N}_{mathcal {G}},mathcal {R}_{mathcal {G}},mathcal {V}_{mathcal {G}} of such holomorphic functions on complete n-circular domain mathcal {G} of mathbb {C}^{n} in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families mathcal {K}_{ mathcal {G}}^{k},kge 2, of holomorphic functions f : mathcal {G}rightarrow mathbb {C},f(0)=1, defined also by a factorization of mathcal {L}f onto factors from mathcal {C}_{mathcal {G}} and mathcal {M} _{mathcal {G}}. We present some interesting properties and extremal problems on mathcal {K}_{mathcal {G}}^{k}.

Algebras of symmetric holomorphic functions of several complex variables

Given a proper holomorphic mapping \(g:\varOmega \subseteq {\mathbb {C}}^{n}\longrightarrow \varOmega ' \subseteq {\mathbb {C}}^{n}\) and an algebra of holomorphic functions \({\mathcal {B}}\) (e.g. \({\mathscr {P}}(K)\) where \(K\subset \varOmega \) is a compact set, \({\mathcal {H}}(U)\), A(U) or \({\mathcal {H}}^{\infty }(U)\) where U is an open and bounded set with \(\overline{U}\subset \varOmega \)), we study the subalgebra \({\mathcal {B}}_{g}\) of all functions compatible with the equivalence relation defined by the proper mapping g. We provide alternative representations of these algebras and describe the fibers in their spectra. Among other examples we relate the algebras of functions that are invariant under permutations and the algebras of functions defined on the symmetrized polydisk.

Computational Problems of Multivariate Hypergeometric Theory

We consider computational problems of the theory of hypergeometric functions in several complex variables: computation of the holonomic rank of a hypergeometric system of partial differential equations, computing the defining polynomial of the singular hypersurface of such a system and finding its monomial solutions. The presented algorithms have been implemented in the computer algebra system MATHEMATICA.