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

Piotr Liczberski,Renata Długosz

doi:10.3390/sym12101707

Abstract

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.

Highlights

Since the several complex variables geometric analysis depends on the type of domains in Cn, we consider a special, but wide class of domains in Cn
ΜG (z) = in f {t > 0 : z ∈ G}, z ∈ Cn. It is well-known that μG is a norm in Cn if G is a convex bounded complete n-circular domain
The function μG is very useful in research the space HG of holomorphic functions f : G → C

Summary

Introduction

Since the several complex variables geometric analysis depends on the type of domains in Cn (see for instance References [1,2,3]), we consider a special, but wide class of domains in Cn. A simple kind of 1-homogeneous polynomial is the following linear functional J ∈ (Cn )∗ We will use the following generalization of the notion of the norm of m-homogeneous polynomial

Results

Conclusion