Biholomorphic Mappings Research Articles

Rigidity of Julia Sets of Families of Biholomorphic Mappings in Higher Dimensions

The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in $$\mathbb C^k$$ , $$k \ge 3.$$ First, we study the relation between two polynomial shift-like maps in $$\mathbb C^k$$ , $$k \ge 3$$ , that share the same forward and backward Julia sets. Secondly, we study the relation between any pair of skew products of Henon maps in $$\mathbb C^3$$ having the same (forward and backward) Julia sets. Also in the same spirit, we further establish a similar relation between skew products of Henon maps fibered over a compact metric space, sharing the same Julia sets.

Horosphere-invariant mappings of the unit ball in [formula omitted

Horospheres, particular ellipsoids internally tangent to the boundary of the Euclidean unit ball B of Cn, play a pivotal role in the Denjoy–Wolff theory of holomorphic self-mappings of B. We use horospheres to introduce a new family of transforms on the class of normalized locally biholomorphic mappings of B into Cn in a vein similar to the Koebe transforms with respect to automorphisms of B. Like the Koebe transforms, these horosphere transforms preserve the property of convexity of a mapping. After studying the transforms with regard to convex mappings in particular, we give a broad characterization of all normalized holomorphic mappings that are invariant under the horosphere transforms. We also show that the notion of horosphere invariance for locally biholomorphic mappings is preserved by several commonly studied extension operators. We then apply our technique to provide a similar, simplified characterization of Koebe-invariant mappings. With these characterizations in hand, we show that, despite the existence of many mappings satisfying either type of invariance separately, the only normalized holomorphic mappings that enjoy both types of invariance are those mappings of the formf(z)=(z11−z1+Q(zˆ)(1−z1)2,zˆ(1−z1)) for z=(z1,zˆ)∈B, where zˆ=(z2,…,zn) and Q:Cn−1→C is a homogeneous polynomial of degree 2. For certain Q, these mappings are extreme points of the closed convex hull of the family of convex mappings of B.

Distortion results for a certain subclass of biholomorphic mappings in ℂ n

Let be the space of n-dimensional complex variables and be the unit polydisc in . We obtain the distortion theorems of the Fréchet-derivative type and the Jacobi-determinant type for a certain subclass of normalized biholomorphic mappings defined on . Also, the distortion theorem of Jacobi-determinant type for the corresponding subclass defined on the unit ball in with arbitrary norm is established. Our results allow each component of complex vectors to have different dimensions, which extends severl previous works being closely related to some subclasses of starlike mappings.

Holomorphic Maps of Levi-Degenerate Tube Hypersurfaces in $$\mathbb C^3$$

Locally biholomorphic maps between 2-nondegenerate smooth real tube hypersurfaces in $$\mathbb C^3$$ with Levi form of rank $$1$$ are described. It is shown that, except for hypersurfaces that are locally equivalent to the boundary of the future tube, such maps must be affine. The proof uses the local holomorphic version of the fundamental theorem of projective geometry which was earlier proved by the author.

Splitting Lemma for Biholomorphic Mappings with Smooth Dependence on Parameters

We prove that the mappings obtained in Forstnerič splitting lemma vary in a mathcal {C}^{lfloor {frac{l-1}{2}}rfloor }-continuous way if only the input family of biholomorphic mappings close to Id (and their domains) is mathcal {C}^l-continuous (see Theorem 1.3 for a precise formulation).

Conformal equivalence of visual metrics in pseudoconvex domains

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between bounded, smooth strongly pseudoconvex domains in {mathbb {C}}^n are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between bounded smooth pseudoconvex domains. The proofs are inspired by Mostow’s proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.

Loewner Chains Associated with Close to Almost Starlike Mappings of Order $$\alpha $$

In this article, we give an equivalent characterization for close to almost starlike mappings of order $$\alpha $$$$(0\le \alpha <1)$$ in terms of Loewner chains. Next, the growth theorem and distortion theorem along a direction are obtained for this subclass of biholomorphic mappings. In particular, the results of Pfaltzgraff and Suffridge can be obtained when $$\alpha =0$$.

The Muir extension operator and Janowski univalent functions

ABSTRACTIn this paper, we continue the work started in [Manu. Extension operators preserving Janowski classes of univalent functions. submitted; 2019] and we shall prove that the Muir extension operator preserves the notion of g-parametric representation, where the function g is given by and . As a consequence, the extension operator preserves the notions of Janowski starlikeness on and Janowski almost starlikeness on . Other particular cases will be mentioned. In the last part of the paper we obtain sharp growth and distortion theorems for some classes of biholomorphic mappings on generated by the extension operator .

Sharp Estimates of Quasi-Convex Mappings of Type B and Order α

In this paper, we first establish several sharp inequalities of homogeneous expansion for biholomorphic quasi-convex mappings of type B and order α on the unit ball E in a complex Banach space X by applying the method and technique of complex analysis. Then, as the application of these sharp inequalities, we derive the sharp estimate of third homogeneous expansions for the above mappings defined on the unit polydisk Un in ℂn.

Loewner chains, Bloch mappings and Pfaltzgraff-Suffridge extension operators on bounded symmetric domains

ABSTRACTLet Y be a complex Banach space and let be the open unit ball of Y. In this paper we consider a generalization of the Pfaltzgraff-Suffridge extension operator on bounded symmetric domains in , and prove that if is a bounded symmetric domain in , and is an extension operator which maps normalized locally biholomorphic mappings on to locally biholomorphic mappings on , where is a certain domain with , then extends the first elements of Loewner chains from to the first elements of Loewner chains on , when , where is a constant defined by the Bergman metric on . Particular cases are also obtained. Next, we prove that normalized locally univalent I-Bloch mappings, which have finite trace order on , are mapped into R-Bloch mappings on by the operator when , where is a bounded balanced convex domain such that . In particular, the normalized bounded convex mappings on are mapped into R-Bloch mappings on under the extension operator .

The Invariance of Subclasses of Biholomorphic Mappings on Bergman-Hartogs Domains

We mainly discuss the invariance of some subclasses of biholomorphic mappings under the generalized Roper-Suffridge operators on Bergman-Hartogs domains which are based on the unit ball Bn. Using the geometric properties and the distortion results of subclasses of biholomorphic mappings, we obtain the geometric characters of almost spirallike mappings of type β and order $$\alpha, S_\Omega^*(\beta, A, B)$$ , strong and almost spirallike mappings of type β and order α maintained under the generalized Roper-Suffridge operators on Bergman-Hartogs domains. Sequentially, we conclude that the generalized operators and the known operators preserve the same properties under some conditions. The conclusions generalize some known results.

Biholomorphic Mappings and the Extension Operators on New Hartogs Domains

In this paper, we generalize the Roper-Suffridge operator on the extended Hartogs domains. By using the geometric properties and the growth theorems of subclasses of biholomorphic mappings, we obtain the generalized operators preserve the properties of parabolic and spirallike mappings of type β and order ρ, $$S_\Omega^*(\beta, A, B)$$ , almost starlike mapping of complex order λ on ΩN under different conditions, and thus we get the corresponding results on the unit ball Bn in ℂn. The conclusions lead to some known results.

Extension Operators for Biholomorphic Mappings

AbstractSuppose that $D\subset \mathbb{C}$ is a simply connected subdomain containing the origin and $f(z_{1})$ is a normalized convex (resp., starlike) function on $D$. Let $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{N}(D)=\bigg\{(z_{1},w_{1},\ldots ,w_{k})\in \mathbb{C}\times \mathbb{C}^{n_{1}}\times \cdots \times \mathbb{C}^{n_{k}}:\Vert w_{1}\Vert _{p_{1}}^{p_{1}}+\cdots +\Vert w_{k}\Vert _{p_{k}}^{p_{k}}&lt;\frac{1}{\unicode[STIX]{x1D706}_{D}(z_{1})}\bigg\},\end{eqnarray}$$ where $p_{j}\geqslant 1$, $N=1+n_{1}+\cdots +n_{k}$, $w_{1}\in \mathbb{C}^{n_{1}},\ldots ,w_{k}\in \mathbb{C}^{n_{k}}$ and $\unicode[STIX]{x1D706}_{D}$ is the density of the hyperbolic metric on $D$. In this paper, we prove that $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)(z_{1},w_{1},\ldots ,w_{k})=(f(z_{1}),(f^{\prime }(z_{1}))^{1/p_{1}}w_{1},\ldots ,(f^{\prime }(z_{1}))^{1/p_{k}}w_{k})\end{eqnarray}$$ is a normalized convex (resp., starlike) mapping on $\unicode[STIX]{x1D6FA}_{N}(D)$. If $D$ is the unit disk, then our result reduces to Gong and Liu via a new method. Moreover, we give a new operator for convex mapping construction on an unbounded domain in $\mathbb{C}^{2}$. Using a geometric approach, we prove that $\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$ is a spiral-like mapping of type $\unicode[STIX]{x1D6FC}$ when $f$ is a spiral-like function of type $\unicode[STIX]{x1D6FC}$ on the unit disk.

Growth and Distortion Results for a Class of Biholomorphic Mapping and Extremal Problem with Parametric Representation in $$\mathbb {C}^n$$ C n

Let $$\widehat{\mathcal {S}}_g^{\alpha , \beta }(\mathbb {B}^n)$$ be a subclass of normalized biholomorphic mappings defined on the unit ball in $$\mathbb {C}^n,$$ which is closely related to the starlike mappings. Firstly, we obtain the growth theorem for $$\widehat{\mathcal {S}}_g^{\alpha , \beta }(\mathbb {B}^n)$$ . Secondly, we apply the growth theorem and a new type of the boundary Schwarz lemma to establish the distortion theorems of the Frechet-derivative type and the Jacobi-determinant type for this subclass, and the distortion theorems with g-starlike mapping (resp. starlike mapping) are partly established also. At last, we study the Kirwan and Pell type results for the compact set of mappings which have g-parametric representation associated with a modified Roper–Suffridge extension operator, which extend some earlier related results.

Subclasses of Biholomorphic Mappings Under the Extension Operators

In this article, we mainly study the invariance of some biholomorphic mappings with special geometric characteristics under the extension operators. First we generalize the Roper-Suffridge extension operators on Bergman-Hartogs domains. Then, by the geometric characteristics of subclasses of biholomorphic mappings, we conclude that the modified Roper-Suffridge operators preserve the properties of SΩ* (β,A,B), parabolic and spirallike mappings of type β and order ρ, strong and almost spirallike mappings of type β and order α as well as almost starlike mappings of complex order λ on $$\Omega_{p_{1}}^{B^{n}},\ldots,_{p_{s},q}$$ under different conditions, respectively. The conclusions provide new approaches to construct these biholomorphic mappings in several complex variables.

Rigidity for convex mappings of Reinhardt domains and its applications

In this paper, we investigate rigidity and its application to extreme points of biholomorphic convex mappings on Reinhardt domains. By introducing a version of the scaling method, we precisely construct many unbounded convex mappings with only one infinite discontinuity on the boundary of this domain. We also give a rigidity of these unbounded convex mappings via Kobayashi metric and Liouville-type theorem of entire functions. As an application we obtain a collection of extreme points for the class of normalized convex mappings. Our results extend both the rigidity of convex mappings and related extreme points from the unit ball to Reinhardt domains.

Extension operators and automorphisms

Many of the commonly studied extension operators that take a normalized locally univalent function f defined in the open unit disk D ⊆ C to a normalized locally biholomorphic mapping of the unit ball B ⊆ C n , n ≥ 2 , are of the form f ↦ Ψ ∘ Φ 0 [ f ] , where Φ 0 [ f ] ( z ) = ( f ( z 1 ) , f ′ ( z 1 ) z ˆ ) , z ∈ B , z ˆ = ( z 2 , … , z n ) , is the Roper–Suffridge extension of f and Ψ is a first-coordinate shear, an automorphism of C n of the form Ψ ( w ) = ( w 1 + g ( w ˆ ) , w ˆ ) , where g : C n − 1 → C is holomorphic and contains no terms of degree 0 or 1 in its Taylor expansion at the origin. Other studies have considered the Roper–Suffridge extension operator with the multiplier f ′ ( z 1 ) replaced by a different expression. Recently, it has been shown that the set of restrictions of automorphisms of C n to B satisfying some geometric property such as convexity or starlikeness is dense in the full respective family of geometric mappings when n ≥ 2 . We study large families of biholomorphic mappings on B generated by automorphisms of C n , abstract generalizations of the Roper–Suffridge extension operator, unitary rotations, and families of locally univalent functions on D . In many cases, families arising from distinct automorphisms will be seen to be disjoint, despite the fact that they contain the restrictions of the respective automorphisms to B and the previously noted density result. Of particular interest given our consideration of more general automorphisms related to extension operators, we show that the compositions of the Roper–Suffridge extension operator with a particular class of first-coordinate overshears, automorphisms of C n of the form Ψ ( w ) = ( h ( w ˆ ) w 1 + g ( w ˆ ) , w ˆ ) for g , h : C n − 1 → C holomorphic and h never 0, extend certain bounded starlike mappings of D to starlike mappings of B . We conclude with several open questions inspired by this work.

A localization principle for biholomorphic mappings between the Fock-Bargmann-Hartogs domains

In this paper, we prove that a localization principle for biholomorphic mappings between equidimensional Fock-Bargmann-Hartogs domains holds. As an application of this, we show that any proper holomorphic mapping between two equidimensional Fock-Bargmann-Hartogs domains satisfying some condition is necessarily a biholomorphic mapping.

Slice Starlike Functions Over Quaternions

In this paper, we initiate the study of the geometric function theory for slice starlike functions over quaternions and its subclasses. This allows us to answer negatively some questions about the Bieberbach conjecture, the growth, distortion, and covering theorems for slice regular functions. Precisely, we find that the Bieberbach conjecture holds true for slice starlike functions in contrast to the fact that the Bieberbach conjecture fails for biholomorphic starlike mappings in higher dimensions. We also establish some sharp versions of the growth, distortion, and covering theorems for quaternions.

Coefficient inequality for a subclass of biholomorphic mappings in several complex variables

Let be the familiar class of starlike functions of order in the unit disk. In Keogh and Merkes proved that for a function in the class , then In this paper, we investigate the corresponding problem for the subclass of starlike mappings of order defined on the unit ball in a complex Banach space, on the unit polydisk in and the bounded starlike circular domain in , respectively.