Abstract The paper is devoted to two new versions of Jack’s lemma for holomorphic functions of several complex variables. The assumption in the first case are formulated in the terms of the real part of certain complex quantities related to Jack’s lemma, and in the second one in the terms of the argument of adequate quantities. All versions are presented common with their applications to the study of Bavrin’s families of s.c.v. holomorphic functions. By Bavrin’s families we mean some collections of holomorphic functions in the open unit polydisc satisfying certain geometric conditions for these functions and their Fréchet differentials of various order.

Distortion theorems for g-parametric quasi-convex mappings in several complex variables

We introduce and study the algebraic, analytic and lattice properties of regular homogeneous polynomials and holomorphic functions on complex Banach lattices. We show that the theory of power series with regular terms is closer to the theory of functions of several complex variables than the theory of holomorphic functions on Banach spaces. We extend the concept of the Bohr radius to Banach lattices and show that it provides us with a lower bound for the ratio between the radius of regular convergence and the radius of convergence of a regular holomorphic function. This allows us to show that in finite dimensions the radius of convergence of the Taylor series of a holomorphic function coincides with the radius of convergence of its monomial expansion but that on ℓ p \ell _p these two radii can be radically different.

This article considers a class of entire functions of several complex variables that are bounded in the Cartesian product of some half-planes. Each such hyperplane is defined on the condition that the real part of the corresponding variable is less than some r. For this class of functions, there are established analogs of the Wiman theorems. The first result describes the behavior of an entire function from the given class at the neighborhood of the point of the supremum of its modulus. The second result shows asymptotic equality for supremums of the modulus of the function and its real part outside some exceptional set. In addition, the analogs of Wiman’s theorem are obtained for entire multiple Dirichlet series with arbitrary non-negative exponents. These results are obtained as consequences of a new statement describing the behavior of an entire function F(z) of several complex variables z=(z1,…,zp) at the neighborhood of a point w, where the value F(w) is close to the supremum of its modulus on the boundary of polylinear domains. The paper has two moments of novelty: the results use a more general geometric exhaustion of p-dimensional complex space by polylinear domains than previously known; another aspect of novelty concerns the results obtained for entire multiple Dirichlet series. There is no restriction that every component of exponents is strictly increasing. These statements are valid for any non-negative exponents.

On the relative growth of entire functions of several complex variables

Systems of nonalgebraic equations containing entire functions of several complex variables are considered. The number of real zeros of such systems is studied using computer algebra methods. For this purpose, a computer implementation of Newton’s recurrences and formulas for the resultant of the functions under study in Maple is proposed. The relevance of this problem is due to the fact that in applied problems, e.g., in the equations of chemical kinetics, it is necessary to determine the number of stationary states of the system.

On the relative growth of entire functions of several complex variables

The paper refers to an aggregated estimate of two initial terms in the power series expansions of holomorphic functions and mappings of several complex variables. The authors solve first this problem in some Bavrin’s families of functions, i.e., functions f:G→C, holomorphic in bounded complete n-circular domains G⊂Cn and satisfying conditions, similar as in geometric function theory of one variable. Next, they apply this result to a family of mappings F:Bn→Cn biholomorphic in the open unit Euclidean ball Bn⊂Cn.

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

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.

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.

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.

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.

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.

We give a survey on recent progress on converses of $$L^2$$ existence theorem and $$L^2$$ extension theorem which are two main parts in $$L^2$$ -theory, and their applications in getting criteria of Griffiths positivity and characterizations of Nakano positivity of (singular) Hermitian metrics of holomorphic vector bundles, as well as the strong openness property and stability property of multiplier submodule sheaves associated to singular Nakano semipositive Hermitian metrics on holomorphic vector bundles.

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.

In this paper, besides a counterexample to Bloch's principle, normality criteria leading to counterexamples to the converse of Bloch's principle in several complex variables are proved.Some Picard-type theorems and their corresponding normality criteria in C n are also obtained.

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

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.

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.