scholarly journals Boundary Schwarz lemma and rigidity property for holomorphic mappings of the unit polydisc in Cn

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2813-2818
Author(s):  
Ziyan Huang ◽  
Di Zhao ◽  
Hongyi Li
Keyword(s):  
Fixed Point ◽  
Unit Disk ◽  
Complex Plane ◽  
Complex Space ◽  
Holomorphic Mappings ◽  
Schwarz Lemma ◽  
Higher Dimensional ◽  
Rigidity Property ◽  
Boundary Schwarz Lemma

In this paper, we generalize the classical Schwarz lemma at the boundary from the unit disk D in the complex plane to the unit polydisc Dn in higher-dimensional complex space. Two boundary Schwarz lemmas for holomorphic mappings of Dn and corresponding rigidity properties are established without the restriction of the interior fixed point.

scholarly journals Commuting semigroups of holomorphic mappings

2008 ◽  
Vol 103 (2) ◽  
pp. 295 ◽  
Author(s):  
Mark Elin ◽  
Marina Levenshtein ◽  
Simeon Reich ◽  
David Shoikhet
Keyword(s):  
Fixed Point ◽  
Unit Disk ◽  
Null Point ◽  
Holomorphic Mappings ◽  
Auxiliary Result ◽  
Derivatives Of

Let $S_{1}=\left\{F_t\right\}_{t\geq 0}$ and $S_{2}=\left\{G_t\right\}_{t\geq 0}$ be two continuous semigroups of holomorphic self-mappings of the unit disk $\Delta=\{z:|z|<1\}$ generated by $f$ and $g$, respectively. We present conditions on the behavior of $f$ (or $g$) in a neighborhood of a fixed point of $S_{1}$ (or $S_{2}$), under which the commutativity of two elements, say, $F_1$ and $G_1$ of the semigroups implies that the semigroups commute, i.e., $F_{t}\circ G_{s}=G_{s}\circ F_{t}$ for all $s,t\geq 0$. As an auxiliary result, we show that the existence of the (angular or unrestricted) $n$-th derivative of the generator $f$ of a semigroup $\left\{F_t\right\}_{t\geq 0}$ at a boundary null point of $f$ implies that the corresponding derivatives of $F_{t}$, $t\geq 0$, also exist, and we obtain formulae connecting them for $n=2,3$.

scholarly journals Schwarz lemma and boundary Schwarz lemma for pluriharmonic mappings

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5385-5402 ◽  
Author(s):  
Jian-Feng Zhu
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Schwarz Lemma ◽  
Classical Boundary ◽  
Boundary Schwarz Lemma

In this paper, we first improve the boundary Schwarz lemma for holomorphic self-mappings of the unit ball Bn, and then we establish the boundary Schwarz lemma for harmonic self-mappings of the unit disk D and pluriharmonic self-mappings of Bn. The results are sharp and coincides with the classical boundary Schwarz lemma when n = 1.

On BMOA for Riemann Surfaces

1981 ◽  
Vol 33 (5) ◽  
pp. 1255-1260 ◽  
Author(s):  
Thomas A. Metzger
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Hardy Class ◽  
Image Position

Let Δ denote the unit disk in the complex plane C. The space BMO has been extensively studied by many authors (see [3] for a good exposition of this topic). Recently, the subspace BMOA (Δ) has become a topic of interest. An analytic function f, in the Hardy class H2(A), belongs to BMOA (Δ) if(1)whereIt is known (see [3, p. 96]) that (1) is equivalent to(2)

scholarly journals Inner composition of analytic mappings on the unit disk

1991 ◽  
Vol 14 (2) ◽  
pp. 221-226 ◽  
Author(s):  
John Gill
Keyword(s):  
Fixed Point ◽  
Unit Disk ◽  
Analytic Functions ◽  
Continued Fraction ◽  
Unique Fixed Point ◽  
Iteration Theory ◽  
Basic Theorem ◽  
Closed Unit Disk ◽  
Compositional Structure ◽  
Compositional Structures

A basic theorem of iteration theory (Henrici [6]) states thatfanalytic on the interior of the closed unit diskDand continuous onDwithInt(D)f(D)carries any pointz ϵ Dto the unique fixed pointα ϵ Doff. That is to say,fn(z)→αasn→∞. In [3] and [5] the author generalized this result in the following way: LetFn(z):=f1∘…∘fn(z). Thenfn→funiformly onDimpliesFn(z)λ, a constant, for allz ϵ D. This kind of compositional structure is a generalization of a limit periodic continued fraction. This paper focuses on the convergence behavior of more general inner compositional structuresf1∘…∘fn(z)where thefj's are analytic onInt(D)and continuous onDwithInt(D)fj(D), but essentially random. Applications include analytic functions defined by this process.

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 Growth Theorems for a Subclass of Strongly Spirallike Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yan-Yan Cui ◽  
Chao-Jun Wang ◽  
Si-Feng Zhu
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Starlike Functions ◽  
Geometric Properties ◽  
Strongly Starlike Functions ◽  
Strongly Starlike ◽  
Spirallike Functions ◽  
Analytic Mappings

In this paper we consider a subclass of strongly spirallike functions on the unit diskDin the complex planeC, namely, strongly almost spirallike functions of typeβand orderα. We obtain the growth results for strongly almost spirallike functions of typeβand orderαon the unit diskDinCby using subordination principles and the geometric properties of analytic mappings. Furthermore we get the growth theorems for strongly almost starlike functions of orderαand strongly starlike functions on the unit diskDofC. These growth results follow the deviation results of these functions.

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