Spirallike Mappings Research Articles

Covering Results and Perturbed Roper–Suffridge Operators

This work is devoted to the advanced study of Roper–Suffridge type extension operators. For a given non-normalized spirallike function (with respect to an interior or boundary point) on the open unit disk of the complex plane, we construct perturbed extension operators in a certain class of Banach spaces and prove that these operators preserve the spirallikeness property. In addition, we present an extension operator for semigroup generators. We use a new geometric approach based on the connection between spirallike mappings and one-parameter continuous semigroups. It turns out that the new one-dimensional covering results established below are crucial for our investigation.

Extension operators via semigroups

The Roper–Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is A-spirallike for a variety of linear operators A. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.

Solution of a Loewner chain equation in several complex variables

We find a solution to the Loewner chain equation in the case when the infinitesimal generator satisfies h ( 0 , t ) = 0 , D h ( 0 , t ) = A for any A ∈ L ( C n , C n ) with m ( A ) > 0 . We also study the related classes of spirallike mappings, mappings with parametric representation and asymptotically spirallike mappings.

On Subordination Chains with Normalization Given by a Time-dependent Linear Operator

In this paper we are concerned with solutions, in particular with univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball Bn in \({\mathbb{C}^n}\). The main result is a generalization to higher dimensions of a well known result due to Becker. Various particular cases of this result have been recently obtained for subordination chains with normalization \({Df(0,t)=e^tI_n}\) or Df(0, t) = etA, t ≥ 0, where \({A\in L(\mathbb{C}^n,\mathbb{C}^n)}\). We also determine the form of the standard solutions to the Loewner differential equation associated with generalized spirallike mappings. In the last section we obtain the form of the solution in the presence of coefficient bounds.

Homeomorphic extension of strongly spirallike mappings in ℂ n

In this paper we consider a proper subclass ŜAn of the full class of spirallike mappings on the Euclidean unit ball Bn in ℂn with respect to a given linear operator A. We use the method of subordination chains to obtain an upper growth result for ŜAn, and we obtain various examples of mappings in the same class of normalized biholomorphic mappings on the unit ball Bn in ℂn. We also prove that the class ŜAn is compact, while the full class of spirallike mappings with respect to a linear operator need not be compact in dimension n ⩾ 2, even when the operator is diagonal. This is one of the motivations for considering the class ŜAn. Finally we prove that if f is a quasiregular strongly spirallike mapping on Bn such that |[Df(z)]−1Af(z)| is uniformly bounded on Bn, then f extends to a homeomorphism of ℝ2n onto itself. In addition, if A+A* = 2aIn for some a > 0, this extension is also quasiconformal on ℝ2n.

Solutions for the generalized Loewner differential equation in several complex variables

We generalize a one-variable result of J. Becker to several complex variables. We determine the form of arbitrary solutions of the Loewner differential equation that is satisfied by univalent subordination chains of the form \({f(z, t)=e^{tA}z+\cdots,}\) where \({A\in L(\mathbb{C}^n, \mathbb{C}^n)}\) has the property k +(A) < 2m(A). Here \({k_+(A)=\max\{\Re\lambda:\lambda\in \sigma(A)\}}\) and \({m(A)=\min\{\Re\langle A(z), z \rangle: \|z\|=1\}}\) . (The notion of parametric representation has a useful generalization under these conditions, so that one has a canonical solution of the Loewner differential equation.) In particular, we determine the form of the univalent solutions. The results are applied to subordination chains generated by spirallike mappings on the unit ball in \({\mathbb{C}^n}\) . Finally, we determine the form of the solutions in the presence of certain coefficient bounds.

Spirallike mappings and univalent subordination chains in Cn

Spirallike mappings and univalent subordination chains in Cn

Parametric representation and asymptotic starlikeness in ℂⁿ

In this paper we consider the notion of asymptotic starlikeness in the Euclidean space C n \mathbb {C}^n . In the case of the maximum norm, asymptotic starlikeness was introduced by Poreda. We have modified his definition slightly, adding a boundedness condition. We prove that the notion of parametric representation which arises in Loewner theory can be characterized in terms of asymptotic starlikeness; i.e. they are equivalent notions. (A regularity assumption of Poreda is not needed.) In particular, starlike mappings and spirallike mappings of type α ∈ ( − π / 2 , π / 2 ) \alpha \in (-\pi /2,\pi /2) are asymptotically starlike. Therefore this notion is a natural generalization of starlikeness. However, we give an example of a spirallike mapping with respect to a linear operator which is not asymptotically starlike. In the case of one complex variable, any function in the class S S is asymptotically starlike; however, in dimension n ≥ 2 n\geq 2 this is no longer true.

Asymptotically spirallike mappings in several complex variables

In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space ℂn. We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric representation, where A ∈ L(ℂn, ℂn), and prove that if \( \int_0^\infty {\parallel e^{(A - 2m(A)I_n )t} \parallel } \)dt < ∞ (this integral is convergent if k+(A) < 2m(A)), then a mapping f ∈ S(Bn) is A-asymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f(z, t) such that D f(0, t) = eAt, {e−Atf(·, t)}t≥0 is a normal family on Bn and f = f(·, 0). In particular, a spirallike mapping with respect to A ∈ L(ℂn, ℂn) with \( \int_0^\infty {\parallel e^{(A - 2m(A)I_n )t} \parallel } \)dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A* = 2In, then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings.

THE PARAMETRIC REPRESENTATION FOR SPIRAL-LIKE MAPPINGS OF TYPE α ON BOUNDED BALANCED PSEUDOCONVEX DOMAINS

Let Ω be a bounded balanced pseudoconvex domain in ℂ n that admits a plurisubhamonic defining function of class C 2. The authors prove that if f is a spiral-like mapping of type α on Ω, then f can be expressed as a limiting form concerning a Schwarz mapping. Moreover, the characteristics for star-like mappings and spiral-like mappings of type α are obtained. As a direct application, the growth theorem for spiral-like mappings of type α is set up.

The growth theorem and quasiconformal extension of strongly spirallike mappings of type α

Let ω be a bounded balanced domain with C 1 plurisubharmonic defining functions in C n. We give the growth theorem for normalized spirallike mappings of type α on ω. We also give a quasiconformal extension of some strongly spirallike mapping of type α on ω.

Rotation estimates and spirals

It is shown that the logarithmic spiral gives the extremum to F. John’s angle distortion problem for plane bilipschitz mappings. The problem of factoring spiral-like mappings into a composition of homeomorphisms with smaller isometric distortion is studied. A space counterpart of the Freedman and He theorem is obtained.

Spirallike non-holomorphic mappings on balanced pseudoconvex domains

In this paper, we will give some conditions of spirallikeness for mappings defined on bounded balanced pseudoconvex domains in Cn with C1 plurisubharmonic defining functions.

Spirallike logharmonic mappings

Univalent logharmonic mappings from the unit disc onto spirallike domains are represented by means of univalent conformal spirallike mappings. Furthermore the class of logharmonic automorphisms on U is completely characterized.

Hull Subordination and Extremal Problems for Starlike and Spirallike Mappings

Hull Subordination and Extremal Problems for Starlike and Spirallike Mappings

Hull subordination and extremal problems for starlike and spirallike mappings

Let F \mathfrak {F} be a compact subset of the family A \mathcal {A} of functions analytic in Δ = { z : | z | &gt; 1 } \Delta = \{ z:\;|z| &gt; 1\} , and let L \mathcal {L} be a continuous linear operator of order zero on A \mathcal {A} . We show that if the extreme points of the closed convex hull of F \mathcal {F} is the set { f 0 ( x z ) } ( | x | = 1 ) \{ {f_0}(xz)\} (|x| = 1) , then L ( f ) \mathcal {L}(f) is hull subordinate to L ( f 0 ) \mathcal {L}({f_0}) in Δ \Delta . This generalizes results of R. M. Robinson corresponding to families F \mathcal {F} of functions that are subordinate to ( 1 + z ) / ( 1 − z ) (1 + z)/(1 - z) or to 1 / ( 1 − z ) 2 1/{(1 - z)^2} . Families F \mathcal {F} to which this theorem applies are discussed and we identify each such operator L \mathcal {L} with a suitable sequence of complex numbers. Suppose that Φ \Phi is a nonconstant entire function and that 0 &gt; | z 0 | &gt; 1 0 &gt; |{z_0}| &gt; 1 . We show that the maximum of Re ⁡ { Φ [ log ⁡ ( f ( z 0 ) / z 0 ) ] } \operatorname {Re} \{ \Phi [\log (f({z_0})/{z_0})]\} over the class of starlike functions of order a is attained only by the functions f ( z ) = z / ( 1 − x z ) 2 − 2 α , | x | = 1 f(z) = z/{(1 - xz)^{2 - 2\alpha }},\;|x| = 1 . A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.