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.

Parametric Representation of Univalent Mappings in Several Complex Variables

2002 ◽  
Vol 54 (2) ◽  
pp. 324-351 ◽  
Author(s):  
Ian Graham ◽  
Hidetaka Hamada ◽  
Gabriela Kohr
Keyword(s):  
Unit Ball ◽  
Existence Theorems ◽  
Parametric Representation ◽  
Positive Real Part ◽  
Complex Variables ◽  
Holomorphic Mappings ◽  
Several Complex Variables ◽  
Positive Real ◽  
Loewner Equation ◽  
Euclidean Unit Ball

AbstractLet B be the unit ball of with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from B into of “positive real part”, is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of B which arises in the study of the Loewner equation, namely the set S0(B) of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr’s work, we consider subsets of S0(B) obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of S0(B) as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in S(B) which can be imbedded in Loewner chains, but which do not have parametric representation.

scholarly journals Foundations of Fatou Theory and a Tribute to the Work of E. M. Stein on Boundary Behavior of Holomorphic Functions

2021 ◽  
Author(s):  
Fausto Di Biase ◽  
Steven G. Krantz
Keyword(s):  
Harmonic Functions ◽  
Boundary Behavior ◽  
Holomorphic Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Several Variables ◽  
The Subject ◽  
Differentiation Of Integrals ◽  
Fatou Theory

AbstractWe lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic functions.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

Remark on Irreducible Factorizations of Polynomials in Several Variables*

1981 ◽  
Vol 24 (4) ◽  
pp. 493-496
Author(s):  
Leiba Rodman
Keyword(s):  
Complex Variables ◽  
Several Complex Variables ◽  
Analytic Set ◽  
Several Variables ◽  
Set Of Points

AbstractIt is proved that a polynomial on several complex variables, whose coefficients depend analytically on a parameter ε, admits a factorization which is irreducible for every value of the parameter, with the possible exception of an analytic set of points. Moreover, the coefficients of the irreducible factors can be chosen to depend analytically on ε in a neighborhood of every point not belonging to this analytic set.

Holomorphic Functions with Positive Real Part

1982 ◽  
Vol 34 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Eric Sawyer
Keyword(s):  
Unit Ball ◽  
Hardy Spaces ◽  
Convex Cone ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Spaces Of Holomorphic Functions ◽  
Extreme Element ◽  
Image Position ◽  
Special Case

The main purpose of this note is to prove a special case of the following conjecture.Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functionsis in Hp(B2) for 0 < p < 3/2.

scholarly journals Research on Geometric Mappings in Complex Systems Analysis

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Yanyan Cui ◽  
Chaojun Wang ◽  
Sifeng Zhu
Keyword(s):  
Banach Spaces ◽  
Systems Analysis ◽  
Starlike Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Coefficient Estimates ◽  
Positive Real ◽  
Reinhardt Domains ◽  
Complex Order ◽  
Starlike Mappings

We mainly discuss the properties of a new subclass of starlike functions, namely, almost starlike functions of complex order λ, in one and several complex variables. We get the growth and distortion results for almost starlike functions of complex order λ. By the properties of functions with positive real parts and considering the zero of order k, we obtain the coefficient estimates for almost starlike functions of complex order λ on D. We also discuss the invariance of almost starlike mappings of complex order λ on Reinhardt domains and on the unit ball B in complex Banach spaces. The conclusions contain and generalize some known results.

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