scholarly journals Extremal Problems of Some Family of Holomorphic Functions of Several Complex Variables

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1304
Author(s):  
Edyta Trybucka
Keyword(s):  
Homogeneous Polynomial ◽  
Type Theorem ◽  
Holomorphic Functions ◽  
Circular Domain ◽  
Extremal Problems ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Sufficient Condition ◽  
Distortion Type

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.

A theorem of continuation for functions of several complex variables

1958 ◽  
Vol 54 (3) ◽  
pp. 377-382 ◽  
Author(s):  
J. G. Taylor
Keyword(s):  
Field Theory ◽  
Circular Domain ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Quantum Field ◽  
Domain Of Holomorphy ◽  
Envelope Of Holomorphy ◽  
Special Cases ◽  
Theory Of Fields ◽  
Holomorphic Continuation

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.

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

2019 ◽  
Vol 16 (2) ◽  
pp. 154-180
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Holomorphic Function ◽  
Logarithmic Derivative ◽  
Holomorphic Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Necessary And Sufficient Condition ◽  
Positive Continuous Function ◽  
Distribution Of Zeros ◽  
Necessary And Sufficient ◽  
Class Of Functions

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.

scholarly journals Analytic Continuation of Diagonals of Laurent Series for Rational Functions

2021 ◽  
pp. 360-368
Author(s):  
Dmitry Yu. Pochekutov
Keyword(s):  
Analytic Continuation ◽  
Laurent Series ◽  
Rational Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Sufficient Condition ◽  
Branch Points ◽  
Gauss Mapping

We describe branch points of complete q-diagonals of Laurent series for rational functions in several complex variables in terms of the logarithmic Gauss mapping. The sufficient condition of non-algebraicity of such a diagonal is proven

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