Abstract
In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .
Highlights
In recent years, analytic functions of several variables with bounded index have been intensively investigated
Let b ∈ Cn \ {0} be a given direction, L : Cn → R+ be a continuous function. Is it possible to replace the condition “F is holomorphic in Cn ” by the condition “F is holomorphic on all slices z0 + tb” and to deduce all known properties of entire functions of bounded L-index in direction for this function class?
The proposed approach can be applied in analytic theory of differential equations
Summary
Analytic functions of several variables with bounded index have been intensively investigated. An entire function F : Cn → C is called a function of bounded L-index in a direction b, if there exists m0 ∈ Z+ such that for every m ∈ Z+ and for all z ∈ Cn one has. ∂kb F (z) = ∂b ∂kb−1 F (z) , k ≥ 2 The least such integer number m0 , obeying (1), is called the L-index in the direction b of the function F and is denoted by Nb ( F, L). Let b ∈ Cn \ {0} be a given direction, L : Cn → R+ be a continuous function Is it possible to replace the condition “F is holomorphic in Cn ” by the condition “F is holomorphic on all slices z0 + tb” and to deduce all known properties of entire functions of bounded L-index in direction for this function class?.
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