Vector-Valued Entire Functions of Several Variables: Some Local Properties

Andriy Ivanovych Bandura,Oleh Bohdanovych Skaskiv,Tetyana Mykhailivna Salo

doi:10.3390/axioms11010031

Andriy Ivanovych Bandura, Oleh Bohdanovych Skaskiv + Show 1 more

Open Access

https://doi.org/10.3390/axioms11010031

Copy DOI

Abstract

The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in join variables, where L:Cn→R+n is a positive continuous function. For vector-valued functions from this class we prove some propositions describing their local properties. In particular, these functions possess the property that maximum of norm for some partial derivative at a skeleton of polydisc does not exceed norm of the derivative at the center of polydisc multiplied by some constant. The converse proposition is also true if the described inequality is satisfied for derivative in each variable.

Highlights

The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in joint variables where L : Cn → R.+ is some positive continuous function
If the components of a Cn -valued bivariate entire function are of bounded index, the function is of bounded index
An entire vector-valued function F : Cn → C p is said to be of bounded L-index, if there exists n0 ∈ Z+ such that for every z ∈ C n and for all J ∈ Zn+ one has

Summary

Introduction

The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in joint variables (see Definition 1 below) where L : Cn → R.+ is some positive continuous function. If the components of a Cn -valued bivariate entire function are of bounded index, the function is of bounded index They presented sufficient conditions providing index boundedness of bivariate vector-valued entire solutions of certain system of partial differential equations with polynomial coefficients. This class of functions is interesting with its connections with value distribution theory [2,3] and analytic theory of differential equation [1,4,5]. Sheremeta [13,14] studied curves of bounded l-index which are analytic in arbitrary bounded domain on a complex plane These mathematicians found sufficient conditions providing l-index boundedness of every analytic solutions for some system of ordinary differential equations. We assume that in future these results will help to study properties of entire vector-valued solutions for system of partial differential equations as in the case of scalar-valued entire functions of several complex variables (see details for the last case in [5])

Notations and Definitions

Local Behavior of Entire Vector-Valued Functions at Skeleton of Polydisc

Conclusions