I Elementary local properties of holomorphic functions of several complex variables

2010 ◽  
pp. 1-19
Author(s):  
Christine Laurent-Thiébaut
Keyword(s):  
Holomorphic Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Local Properties

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 Some Results of Fekete-Szegö Type. Results for Some Holomorphic Functions of Several Complex Variables

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1707
Author(s):  
Renata Długosz ◽  
Piotr Liczberski
Keyword(s):  
Power Series ◽  
Power Series Expansion ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Homogeneous Polynomials ◽  
Positive Real ◽  
Sharp Estimates ◽  
Several Variables

This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform Lf has positive real part, respectively. The main result arise some sharp estimates of the Minkowski balance of a combination of 2-homogeneous and the square of 1-homogeneous polynomials occurred in power series expansion of functions from aforementioned families.

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