Several Complex Variables Research Articles

The Generalized Toeplitz Determinants for a Class of Holomorphic Mappings in Several Complex Variables

The Generalized Toeplitz Determinants for a Class of Holomorphic Mappings in Several Complex Variables

Homogeneous Projective Coordinates for the Bondi–Metzner–Sachs Group

This paper studies the Bondi–Metzner–Sachs group in homogeneous projective coordinates because it is then possible to write all transformations of such a group in a manifestly linear way. The 2-sphere metric, the Bondi–Metzner–Sachs metric, asymptotic Killing vectors, generators of supertranslations as well as boosts and rotations of Minkowski spacetime are all re-expressed in homogeneous projective coordinates. Lastly, the integral curves of vector fields which generate supertranslations are evaluated in detail. This work paves the way for more advanced applications of the geometry of asymptotically flat spacetime in projective coordinates by virtue of the tools provided from complex analysis in several variables and projective geometry.

On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations

This paper has involved the use of a variety of variations of the Fermat-type equation $f^n(z)+g^n(z)=1$, where $n(\geq 2)\in\mathbb{N}$. Many researchers have demonstrated a keen interest to investigate the Fermat-type equations for entire and meromorphic solutions of several complex variables over the past two decades. Researchers utilize the Nevanlinna theory as the key tool for their investigations. Throughout the paper, we call the pair $(f,g)$ as a finite order entire solution for the Fermat-type compatible system $\begin{cases} f^{m_1}+g^{n_1}=1;\\ f^{m_2}+g^{n_2}=1,\end{cases}$\!\! if $f$, $g$ are finite order entire functions satisfying the system, where $m_1,m_2,n_1,n_2\in\mathbb{N}\setminus\{1\} .$\ Taking into the account the idea of the quadratic trinomial equations, a new system of quadratic trinomial equations has been constructed as follows: $\begin{cases} f^{m_1}+2\alpha f g+g^{n_1}=1;\\ f^{m_2}+2\alpha f g+g^{n_2}=1,\end{cases}$ \!\! where $\alpha\in\mathbb{C}\setminus\{0,\pm1\}.$ In this paper, we consider some earlier systems of certain Fermat-type partial differential-difference equations on $\mathbb{C}^2$, especially, those of Xu {\it{et al.}} (Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483(2), 2020) and then construct some systems of certain quadratic trinomial partial differential-difference equations with arbitrary coefficients. Our objective is to investigate the forms of the finite order transcendental entire functions of several complex variables satisfying the systems of certain quadratic trinomial partial differential-difference equations on $\mathbb{C}^n$. These results will extend the further study of this direction.

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn’s hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm.

Recent progress in the theory of functions of several complex variables and complex geometry

Recent progress in the theory of functions of several complex variables and complex geometry

On Grothendieck-type duality for spaces of holomorphic functions of several variables

We describe the strong dual space $({\mathcal O} (D))^*$ of the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables in a bounded domain $D$ with Lipschitz boundary and connected complement (as usual, ${\mathcal O} (D)$ is endowed with the topology of local uniform convergence in $D$). We identify the dual space with the closed subspace of the space of harmonic functions on the closed set ${\mathbb C}^n\setminus D$, $n>1$, whose elements vanish at the point at infinity and satisfy the Cauchy-Riemann tangential conditions on $\partial D$. In particular, we generalize classical Grothendieck-Köthe-Sebastião e Silva duality for holomorphic functions of one variable to the multivariate situation. We prove that the duality we produce holds if and only if the space ${\mathcal O} (D)\cap H^1 (D)$ of Sobolev-class holomorphic functions in $D$ is dense in ${\mathcal O} (D)$. Bibliography: 35 titles.

Recent progress in the theory of functions of several complex variables and complex geometry

Дан обзор недавних новых результатов в области обратных теорем о $L^2$-существовании и $L^2$-продолжении, составляющих две основные части $L^2$-теории. Эти результаты используются для получения критерия положительности по Гриффитсу и условий положительности по Накано для (сингулярных) эрмитовых метрик голоморфных векторных расслоений, а также для доказательства сильной открытости и устойчивости пучков мультипликативных подмодулей, связанных с сингулярными неотрицательными по Накано эрмитовыми метриками на голоморфных векторных расслоениях.

On Polyanalytic Functions in Several Complex Variables

We give a characterisation of the polyanalytic type subspaces of the Hilbert spaces mathcal {H}, being the weighted L_2 function spaces on a connected simply connected domains D subset mathbb {C}^n. The typical examples considered in the paper are the unit ball mathbb {B}^n and the whole space mathbb {C}^n. Our approach is based on the use of the two tuples of operators a=(a1,a2,…,an)andb=(b1,b2,…,bn),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathbf {\\mathfrak {a}}= (\\mathfrak {a}_1, \\ \\mathfrak {a}_2, \\ \\ldots , \\ \\mathfrak {a}_n) \\quad \ extrm{and} \\quad \\mathbf {\\mathfrak {b}}= (\\mathfrak {b}_1, \\ \\mathfrak {b}_2, \\ \\ldots , \\ \\mathfrak {b}_n), \\end{aligned}$$\\end{document}which act invariantly in some linear space and satisfy therein the commutation relations aj,bℓ=δj,ℓI,aj,aℓ=0,bj,bℓ=0,j,ℓ=1,2,...,n.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left[ \\mathfrak {a}_j, \\mathfrak {b}_{\\ell }\\right] = \\delta _{j,\\ell }I, \\quad \\left[ \\mathfrak {a}_j, \\mathfrak {a}_{\\ell }\\right] = 0, \\quad \\left[ \\mathfrak {b}_j, \\mathfrak {b}_{\\ell }\\right] = 0, \\quad j,\\ell = 1,2,...,n. \\end{aligned}$$\\end{document}We assume further that a common invariant domain of the above operators is a dense linear subspace in a Hilbert space mathcal {H}, and impose several additional conditions. The exact formulation is given in the Extended Fock space construction. Further we specify the obtained description to different concrete realizations of the above operators and Hilbert spaces mathcal {H}, illustrating a variety of possibilities that may occur in the characterisation of the polyanalytic type spaces in several complex variables.

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

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

SOME RESULTS OF THE GROWTH PROPERTIES OF ENTIRE FUNCTIONS OF SEVERAL COMPLEX VARIABLES ON THE BASIS OF THEIR (p, q)th RELATIVE GOL'DBERG ORDER AND (p, q)th-RELATIVE GOL'DBERG TYPE

Biswas [2] introduced the idea of (p; q)th-relative Gol'dberg order and (p; q)th-relative Gol'dberg type of an entire function of several complex variables. In this paper we want to establish some results of the growth analysis of entire function of several complex variables on the basis of their (p, q)-Ψ relative Gol'dberg order and (p, q)-Ψ relative Gol'dberg type of an entire function of several complex variables.

Generalization of Lohwater-Pommerenke's theorem

In this paper, as an application of Zalcman's lemma in ℂn, we give a sufficient condition for normality of holomorphic functions of several complex variables, which generalizes previous known one-dimensional criterion of A.J. Lohwater and Ch. Pommerenke.

GROWTH PROPERTIES OF AN ENTIRE FUNCTION OF SEVERAL COMPLEX VARIABLES ON THE BASIS OF RELATIVE ORDER

In this paper, we study some comparative growth properties of composite entire function of several complex variables, on the basis of relative order and relative lower order of an entire function with respect to an entire function. Here we are defining some definitions related to relative order and relative lower order in terms of central index. A Rastogi, Biswas, Pramanik, Somsundaram many authors work on central index.

Topological properties of the Fréchet space of holomorphic functions of several complex variables

Let 𝑁p(𝐵n) (𝑝 > 1) be the Privalov class of holomorphic functions on the unit ball 𝐵n in the space of 𝑛-complex variables. The class 𝑁p(𝐵n) (𝑝 > 1), equipped with the topology given by a natural metric, becomes an 𝐹-algebra. In this paper, we shall introduce a Fréchet space Fp(𝐵n) (𝑝 > 1) of holomorphic functions on 𝐵n which contains 𝑁p(𝐵n). Moreover, we shall characterize some topological properties of Fp(𝐵n) induced by the family of semi norms on Fp(𝐵n) .

Problem of Generators in a Single Ring of Entire Functions of Several Complex Variables

Problem of Generators in a Single Ring of Entire Functions of Several Complex Variables

New Subclasses of Univalent Mappings in Several Complex Variables: Extension Operators and Applications

New Subclasses of Univalent Mappings in Several Complex Variables: Extension Operators and Applications

Boundedness of the L-index in a direction of the sum and product of slice holomorphic functions in the unit ball

Let $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ with the unit ball $\mathbb{B}^n=\{z\in\mathbb{C}^{n}: \ |z|:=\sqrt{|z|_1^2+\ldots+|z_n|^2}<1\}$ for any $z^0\in\mathbb{B}^n$. For this class of functions there is considered the concept of boundedness of $L$-index in the direction $\mathbf{b},$ where ${L}: \mathbb{B}^n\to\mathbb{R}_+$ is a positive continuous function such that $L(z)>\frac{\beta|\mathbf{b}|}{1-|z|}$ and $\beta>1$ is some constant.There are presented sufficient conditions that the sum of slice holomorphic functions of bounded $L$-index in direction belong this class. This class of slice holomorphic functions is closed under the operation of multiplication.

Slice regular functions in several variables

In this paper, we lay the foundations of the theory of slice regular functions in several (non-commuting) variables ranging in any real alternative ^*-algebra, including quaternions, octonions and Clifford algebras. This higher dimensional function theory is an extension of the classical theory of holomorphic functions of several complex variables. It is based on the construction of a family of commuting complex structures on {mathbb {R}}^{2^n}. One of the relevant aspects of the theory is the validity of a Cauchy-type integral formula and the existence of ordered power series expansions. The theory includes all polynomials and power series with ordered variables and right coefficients in the algebra. We study the real dimension of the zero set of polynomials in the quaternionic and octonionic cases and give some results about the zero set of polynomials with Clifford coefficients. In particular, we show that a nonconstant polynomial always has a non empty zero set.

Composition of entire and analytic functions in the unit ball

In this paper, we investigate a composition of entire function of several complex variables and analytic function in the unit ball. We modified early known results with conditions providing equivalence of boundedness of $L$-index in a direction for such a composition and boundedness of $l$-index of initial function of one variable, where the continuous function $L:\mathbb{B}^n\to \mathbb{R}_+$ is constructed by the continuous function $l: \mathbb{C}^m\to \mathbb{R}_+.$ Taking into account new ideas from recent results on composition of entire functions, we remove a condition that a directional derivative of the inner function $\Phi$ in the composition does not equal to zero. Instead of the condition we construct a greater function $L(z)$ for which $F(z)=f(\underbrace{\Phi(z),\ldots,\Phi(z)}_{m\text{ times}})$ has bounded $L$-index in a direction, where $f\colon \mathbb{C}^m\to \mathbb{C}$ is an entire function of bounded $l$-index in the direction $(1,\ldots,1)$, $\Phi\colon \mathbb{B}^n\to \mathbb{C}$ is an analytic function in the unit ball.
 We weaken the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K|\partial_{\mathbf{b}}\Phi(z)|^k$ for all $z\in\mathbb{B}^n$, where $K\geq 1$ is a constant, $\mathbf{b}\in\mathbb{C}^n\setminus\{0\}$ is a given direction and $${\partial_{\mathbf{b}} F(z)}:=\sum\limits_{j=1}^{n}\!\frac{\partial F(z)}{\partial z_{j}}{b_{j}}, \ \partial_{\mathbf{b}}^k F(z):=\partial_{\mathbf{b}}\big(\partial_{\mathbf{b}}^{k-1} F(z)\big).$$ It is replaced by the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K(l(\Phi(z)))^{1/(N_{\mathbf{1}}(f,l)+1)}|\partial_{\mathbf{b}}\Phi(z)|^k$, where $N_{\mathbf{1}}(f,l)$ is the $l$-index of the function $f$ in the direction $\mathbf{1}=(1,\ldots,1).$ The described result is an improvement of previous one. It is also a new result for the one-dimensional case $n=1,$ $m=1$, i.e. for an analytic function $\Phi$ in the unit disc and for an entire function $f: \mathbb{C}\to\mathbb{C}$ of bounded $l$-index.

L2 estimates and existence theorems for the [formula omitted] operators in infinite dimensions, I

The classical L2 method for the ∂‾ operators is a basic tool in complex analysis of several variables. Naturally, it is expected to extend this method to infinite dimensional complex analysis, but this is a longstanding unsolved problem, due to the essential difficulty that there exists no nontrivial translation invariance measure in the setting of infinite dimensions. The main purpose in this series of work is to give an affirmative solution to the above problem. In this first part, we focus on the simplest case, i.e., L2 estimates and existence theorems for the ∂‾ equations on the whole space of ℓp for p∈[1,∞). The key of our approach is to introduce a suitable working space, i.e., a Hilbert space for (s,t)-forms on ℓp (for each nonnegative integers s and t), via which we define the ∂‾ operator from (s,t)-forms to (s,t+1)-forms and establish the exactness of these operators, and therefore in this case we solve a problem which has been open for forty years.

Note on the Fekete–Szegö Problem for Spirallike Mappings in Banach Spaces

In this note we present a remark on the paper “On the coefficient inequalities for a class of holomorphic mappings associated with spirallike mappings in several complex variables” by Y. Lai and Q. Xu [10] published recently in the journal Results in Mathematics. We show that one of the theorems in [10] concerning the finite-dimensional space \({{\mathbb {C}}}^n\) is a direct consequence of another one, so it does not need an independent proof. Moreover, we prove that a sharp norm estimate on the Fekete–Szegö functional over spirallike mappings in a general Banach space can be deduced from a result in [10].