Biholomorphic Convex Mappings Research Articles

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.

On the Biholomorphic Convex Mappings of Order Alpha on $$D_p^n$$ D p n

In this paper, we first give the definition of biholomorphic convex mappings of order \(\alpha \) on the Reinhardt domain \(D_{p}^n\) in \(C^{n}\). Next, we provide several sufficient conditions for it. Then we introduce a subclass \(SK(D_{p}^n, \alpha )\) of biholomorphic convex mappings of order \(\alpha \) on \(D_{p}^n\), and give a necessary and sufficient condition for the subclass \(SK(D_{p}^n, \alpha )\). From these, we construct some concrete examples of biholomorphic convex mappings of order \(\alpha \) on \(D_{p}^n\). The results presented extend the related results of earlier authors.

Construction of Biholomorphic Convex Mappings of OrderαonBpn

Some sufficient conditions for biholomorphic convex mappings of orderαon the Reinhardt domainBpninCnare given; from that, criteria for biholomorphic convex mappings of orderαwith particular form become direct. As applications of these sufficient conditions, some concrete biholomorphic convex mappings of orderα onBpnare provided.

Biholomorphic convex mappings of order α on

In this article, we shall prove some sufficient conditions for biholomorphic convex mappings of order α on the Reinhardt domain in ℂ n . Some results of Roper and Suffridge, Hamada and Kohr on are extended to . From these, we may construct some concrete biholomorphic convex mappings of order α on . Moreover, an Alexander type theorem between the subclass of convex mappings and the subclass of starlike mappings on is provided.

Biholomorphic convex mappings of order [formula omitted] on the unit ball in Hilbert spaces

In this paper, we introduce the concept of biholomorphic convex mapping of order α on the unit ball B in a complex Hilbert space X. Then we provide some sufficient conditions that a locally biholomorphic mapping f is a biholomorphic convex mapping of order α; we give an Alexander type theorem between the subclass of convex mappings and the subclass of starlike mappings on B in a Hilbert space. We also study the order of starlikeness of biholomorphic convex mappings of order α on B in a Hilbert space. Finally, we construct some examples of biholomorphic convex mappings of order α on B in a Hilbert space by a linear operator.

Construction of Biholomorphic Convex Mappings on $D_p$ IN $C^n$

Construction of Biholomorphic Convex Mappings on $D_p$ IN $C^n$

Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

In this paper, we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball B in a complex Hilbert space X is a biholomorphic convex mapping, which improves some results of Hamada and Kohr and solves the problem which is posed by Graham and Kohr. From this, we derive some sufficient conditions for biholomorphic convex mapping. We also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on B in Hilbert spaces. Moreover, we give some examples of biholomorphic convex mappings on B in Hilbert spaces.

Some sufficient conditions for biholomorphic convex mappings on [formula omitted

A few biholomorphic convex mappings on the unit ball in C n for n > 1 are known. In this paper, we shall prove some sufficient conditions for biholomorphic convex mappings on Reinhardt domain B p n in C n . Some results of Roper and Suffridge, Hamada and Kohr on B 2 n are extended to B p n . From these, we may construct some concrete biholomorphic convex mappings on B p n .

Distortion theorems for biholomorphic convex mappings in Banach spaces*

In this article, we give some distortion theorems for biholomorphic convex mappings in Banach spaces and Hilbert spaces. In particular, we prove that the conjecture of Hamada and Kohr is true in Banach spaces.

On Decomposition Theorem of normalized biholomorphic convex mappings in Reinhardt domains

The construction of normalized biholomorphic convex mappings in the Reinhardt domain \(D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| {z_n } \right|^{p_n } 2, j = 1,2,⋯, n) of ℂn is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component fj, j = 1, 2, ⋯, n, depends only on zj, where k is the number that satisfies k < min {p1, p2,⋯, pn ≤ k + 1. When p1,p2, ... ,pn → ∞ , this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.

Distortion theorems for biholomorphic convex mappings in [formula omitted

In the paper the problem of sharp lower estimation for ‖Df(z)‖ in the class of normalized biholomorphic mappings f between the open unit ball B n and convex domains in C n has been considered.

DISTORTION THEOREMS FOR BIHOLOMORPHIC CONVEX MAPPINGS ON BOUNDED CONVEX CIRCULAR DOMAINS

In terms of Caratheodory metric and Kobayashi metric, distortion theorems for biholomorphic convex mappings on bounded circular convex domains are given.

Decomposition theorem of normalized biholomorphic convex mappings

Decomposition theorem of normalized biholomorphic convex mappings

Growth theorem of convex mappings on bounded convex circular domains

The growth theorem and the 1/2-covering theorem are obtained for the class of normalized biholomorphic convex mappings on bounded convex circular domains, which extend the corresponding results of Sufridge, Thomas, Liu, Gong, Yu, and Wang. The approach is new, which does not appeal to the automorphisms of the domains; and the domains discussed are rather general on which convex mappings can be studied, since the domain may not have a convex mapping if it is not convex.

Homogeneous expansions of normalized biholomorphic convex mappings overB P

The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain $$B^p = \left\{ z \right. \in \mathbb{C}^n :\left\| { z } \right\| _p = [\mathop \sum \limits_{j = 1} \left| {Z_j } \right| ^p ]^{1/p} 2 )$$ are studied. It is proved that the first (k+1) terms of the expansions of the jth componentf j of such a mapf depend only onz j , for 1 ⩽j⩽n, wherek is the natural number that satisfiesk < ρ ⩽k +I. Whenp→ ∞, this gives the result on the unit polydisc obtained by Suffridge in 1970.

Biholomorphic convex mappings of ball in ℂn

Biholomorphic convex mappings of ball in ℂ^<i>n</i>