Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂ n *

2007 ◽  
Vol 28 (6) ◽  
pp. 677-684 ◽  
Author(s):  
Jianfei Wang ◽  
Taishun Liu
Keyword(s):  
Unit Ball ◽  
Holomorphic Mappings ◽  
Bloch Constant

scholarly journals SCHWARZ LEMMA FOR HOLOMORPHIC MAPPINGS IN THE UNIT BALL

2017 ◽  
Vol 60 (1) ◽  
pp. 219-224 ◽  
Author(s):  
DAVID KALAJ
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Type Inequality ◽  
Holomorphic Mappings ◽  
Schwarz Lemma ◽  
Complex Spaces

AbstractIn this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn and Bm in corresponding complex spaces. We also prove a Schwarz-Pick type inequality for pluri-harmonic functions.

scholarly journals Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Jianfei Wang
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Arbitrary Order ◽  
Complex Banach Space ◽  
Holomorphic Mappings ◽  
Partial Derivatives ◽  
Carathéodory Metric ◽  
Homogeneous Ball ◽  
Derivatives Of

LetBXbe the unit ball in a complex Banach spaceX. AssumeBXis homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ballBntoBXassociated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

Holomorphic Mappings of the Hyperbolic Space into the Complex Euclidean Space and the Bloch Theorem

1975 ◽  
Vol 27 (2) ◽  
pp. 446-458 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Hyperbolic Space ◽  
Holomorphic Mapping ◽  
Continuous Functions ◽  
Holomorphic Mappings ◽  
Euclidean Metric ◽  
The Family ◽  
Complex Euclidean Space ◽  
Bounded Holomorphic

This paper is to study various properties of holomorphic mappings defined on the unit ball B in the complex euclidean space Cn with ranges in the space Cm. Furnishing B with the standard invariant Kähler metric and Cm with the ordinary euclidean metric, we define, for each holomorphic mapping f : B → Cm, a pair of non-negative continuous functions qf and Qf on B ; see § 2 for the definition.Let (Ω), Ω > 0, be the family of holomorphic mappings f : B → Cn such that Qf(z) ≦ Ω for all z ∈ B. (Ω) contains the family (M) of bounded holomorphic mappings as a proper subfamily for a suitable M > 0.

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 ON HARMONIC BLOCH SPACES IN THE UNIT BALL OF ℂn

2011 ◽  
Vol 84 (1) ◽  
pp. 67-78 ◽  
Author(s):  
SH. CHEN ◽  
X. WANG
Keyword(s):  
Unit Ball ◽  
Lipschitz Functions ◽  
Harmonic Mappings ◽  
Bloch Spaces ◽  
Bloch Constant ◽  
Vector Valued

AbstractIn this paper, our main aim is to discuss the properties of harmonic mappings in the unit ball 𝔹n. First, we characterize the harmonic Bloch spaces and the little harmonic Bloch spaces from 𝔹n to ℂ in terms of weighted Lipschitz functions. Then we prove the existence of a Landau–Bloch constant for a class of vector-valued harmonic Bloch mappings from 𝔹n to ℂn.

scholarly journals Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jorge J. Garcés ◽  
Antonio M. Peralta ◽  
Daniele Puglisi ◽  
María Isabel Ramírez
Keyword(s):  
Unit Ball ◽  
Taylor Series ◽  
Holomorphic Mapping ◽  
Invertible Element ◽  
Holomorphic Mappings ◽  
Open Unit ◽  
Holomorphic Maps ◽  
Open Unit Ball ◽  
C Algebras ◽  
Orthogonally Additive

We study holomorphic maps between C*-algebrasAandB, whenf:BA(0,ϱ)→Bis a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ballU=BA(0,δ). If we assume thatfis orthogonality preserving and orthogonally additive onAsa∩Uandf(U)contains an invertible element inB, then there exist a sequence(hn)inB**and Jordan*-homomorphismsΘ,Θ~:M(A)→B**such thatf(x)=∑n=1∞hnΘ~(an)=∑n=1∞Θ(an)hnuniformly ina∈U. WhenBis abelian, the hypothesis ofBbeing unital andf(U)∩inv(B)≠∅can be relaxed to get the same statement.

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