Holomorphic Functions and Integral Representations in Several Complex Variables

1986 ◽  
Author(s):  
R. Michael Range
Keyword(s):  
Holomorphic Functions ◽  
Complex Variables ◽  
Integral Representations ◽  
Several Complex Variables

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.

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