Geometric Function Theory in One and Higher Dimensions

The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods. The intention of Gunning and Rossi's book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces. Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces.

Many authors, e.g., Bavrin, Jakubowski, Liczberski, Pfaltzgraff, Sitarski, Suffridge, and Stankiewicz, have discussed some families of holomorphic functions of several complex variables described by some geometrical or analytical conditions. We consider a family of holomorphic functions of several complex variables described in n-circular domain of the space C n . We investigate relations between this family and some of type of Bavrin’s families. We give estimates of G-balance of k-homogeneous polynomial, a distortion type theorem and a sufficient condition for functions belonging to this family. Furthermore, we present some examples of functions from the considered class.

Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of
 topological bases etc. With an idea of correlating entire functions with certain aspects in the theory of basis in locally convex spaces, we have investigated in this paper the bornological aspects of the space $X$ of integral functions of several complex variables. By $Y$ we denote the space of all power series with positive radius of convergence at the origin. We introduce bornologies on $X$
 and $Y$ and prove that $Y$ is a convex bornological vector space which is the completion of the convex bornological vector space $X$.

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.

1. Some problems arising in the study of value distribution of functions of two complex variables. One of the objectives of modern analysis consists in the generalization of methods of the theory of functions of one complex variable in such a way that the procedures in the revised form can be applied in other fields, in particular, in the theory of functions of several complex variables, in the theory of partial differential equations, in differential geometry, etc. In this way one can hope to obtain in time a unified theory of various chapters of analysis. The method of the kernel function is one of the tools of this kind. In particular, this method permits us to develop some chapters of the theory of analytic and meromorphic functions f(z1, * * *, zn) of the class j2(I2n), various chapters in the theory of pseudo-conformal transformations (i.e., of transformations of the domains V2n by n analytic functions of n complex variables) etc. On the other hand, it is of considerable interest to generalize other chapters of the theory of functions of one variable, at first to the case of several complex variables. In particular, the study of value distribution of entire and meromorphic functions represents a topic of great interest. Generalizing the classical results about the zeros of a polynomial, Hadamard and Borel established a connection between the value distribution of a function and its growth. A further step of basic importance has been made by Nevanlinna and Ahlfors, who showed not only that the results of Hadamard and Borel in a sharper form can be obtained by using potential-theoretical and topological methods, but found in this way important new relations, and opened a new field in the modern theory of functions. As one passes from the theory of functions of one variable to the case of two and more variables(2), the question of the value distribution becomes more complex and many directions arise which can be considered as a generalization of the above mentioned chapter of the theory of functions of one variable. However, the formulation of the problems which can be answered, and generalization of methods used in one variable, is not immediate. A function of two complex variables may vanish in a domain on a segment

In this paper, we extend Ahlfors' theory (see W.K. HaymanMeromorphic Functions, Oxford, Clarendon Press, 1964) to quasimeromorphic mappings of several complex variables. From an improvement of Ahlfors' Theorem on the sphere, which gives an explicit estimation of the constant in the theorem, we get the fundamental inequality of the value distribution of quasimeromorphic mappings of 1 -variable. Furthermore, we give a normality criteria for quasimeromorphic mapping family of several complex variables.

The principal objective of this paper is to introduce the ideas of relative $\varphi $-type, relative $\varphi $-weak type of entire functions of several complex variables and study some growth properties concerning them.

We prove a realization theorem for rational functions of several complex variables which extends the main theorem of Bessmertnyĭ (in: Alpay, Gohberg, Vinnikov (eds) Interpolation Theory, Systems Theory and Related Topics. Operator Theory Advances and Applications vol 134. Birkhauser Verlag, Basel, pp 157–185, 2002). In contrast to Bessmertnyĭ’s approach of solving large systems of linear equations, we use an operator theoretical approach based on the theory of Schur complements. This leads to a simpler and more “natural” construction to solving the realization problem as we need only apply elementary algebraic operations to Schur complements such as sums, products, inverses, and compositions. A novelty of our approach is the use of Kronecker product as opposed to the matrix product in the realization problem. As such our synthetic approach leads to a solution of the realization problem that has potential for further extensions and applications within multidimensional systems theory especially for those linear models associated with electric circuits, networks, and composites.

In this chapter, we develop some growth properties of entire functions of n complex variables relating to generalized relative Gol`dberg order (α,&#946); generalized rel- ative Gol`dberg type (α,&#946) and generalized relative Gol`dberg weak type (α,&#946):We also establish integral representations of generalized relative Gol`dberg type and weak type (α,&#946) of entire function of several complex variables and derive some interesting results, where α,&#946 are continuous non-negative functions defined on (-∞,+∞).

Comparisons of ideal structures in algebras of analytic functions of several complex variables

This paper presents results on some boundary value problems for holomorphic functions of several complex variables in polydomains. The Cauchy kernel is one of the significant tools for solving the Riemann and the Riemann-Hilbert boundary value problems for holomorphic functions as well as for establishment of the connection between them. For polydomains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in polydomains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. The general integral representation formulas for the functions, holomorphic in polydomains, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary polydomains is given and an exact, yet compact way of notation for holomorphic functions in arbitrary polydomains is introduced and applied. The Riemann jump problem and the Riemann-Hilbert problem are solved for holomorphic functions of several complex variables with the unit torus as the jump manifold. The higher-dimensional Plemelj-Sokhotzki formula for holomorphic functions in polydomains is established. The canonical functions of the Riemann problem for polydomains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. For all three boundary value problems, well-posed formulations are given which does not demand more solvability conditions than in the one variable case. The connection between the Riemann and the Riemann-Hilbert problem for polydomains is proven. Thus contrary to earlier research the results are similar to the respective ones for just one variable.

We propose a new method to solve the Riemann problem for analytic functions of several complex variables. The essential idea is explained in detail.Key wordsSeveral complex variablesanalytic functionsboundary value problemsolvability condition

Univalent Mappings in Several Complex Variables

It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to impose two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SCDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk.

In this paper, the sharp growth, covering theorems and the sharp distortion theorems of the Fréchet derivative type for a subclass of quasi-convex mappings (include quasi-convex mappings of type and quasi-convex mappings of type ) on the unit ball of complex Banach spaces are first established. Meanwhile, the sharp growth, covering theorems and the sharp distortion theorems of the Fréchet derivative type for the above generalized mappings on the unit polydisc in are given. Moreover, the refined distortion theorems of the Jacobi determinant type for a subclass of quasi-convex mappings (include quasi-convex mappings of type ) on the unit ball with arbitrary norm in are also obtained, and the refined distortion theorems of the Jacobi determinant type for the above generalized mappings on the unit polydisc in are derived. Our results generalize some known conclusions and reduce to the classical results in one complex variable.