The growth and 1∕4-theorems for starlike mappings in Cn

Barnard, Roger W.; Fitzgerald, Carl H.; Gong, Shuxi

doi:10.2140/pjm.1991.150.13

Cited by 39 publications

(22 citation statements)

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“…. , p n > 1, we can show that the estimates are sharp as in Theorem 2.1 of Barnard, FitzGerald and Gong [1].…”

Section: Remark 1 (I) It Is Mentioned In Gong Wang and Yumentioning

confidence: 51%

“…Barnard, FitzGerald and Gong [1] and Chuaqui [2] extended this to normalized starlike mappings on the unit ball B n in C n . Their proof uses the characterization of the starlikeness due to Suffridge [11].…”

Section: Introductionmentioning

confidence: 91%

“…In this section, we give the growth and 1/4-theorems for normalized starlike mappings on bounded balanced pseudoconvex domains with C 1 plurisubharmonic defining functions using the ideas of Barnard, FitzGerald and Gong [1] and Chuaqui [2]. A holomorphic mapping f is said to be normalized if f (0) = 0 and Df (0) = I.…”

Section: Remark 1 (I) It Is Mentioned In Gong Wang and Yumentioning

confidence: 99%

“…We say that f has a k-fold symmetric image if the image of f is unchanged when multiplied by the scalar complex number exp(2πi/k). If k-fold symmetry of f is assumed, then Theorems 4, 5 and Corollary 1 can be strengthened as follows as in Barnard, FitzGerald and Gong [1].…”

Section: Remark 1 (I) It Is Mentioned In Gong Wang and Yumentioning

confidence: 99%

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Starlike mappings on bounded balanced domains with C¹-plurisubharmonic defining functions

Hamada¹

2000

Pacific J. Math.

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“…. , p n > 1, we can show that the estimates are sharp as in Theorem 2.1 of Barnard, FitzGerald and Gong [1].…”

Section: Remark 1 (I) It Is Mentioned In Gong Wang and Yumentioning

confidence: 51%

Section: Introductionmentioning

confidence: 91%

Section: Remark 1 (I) It Is Mentioned In Gong Wang and Yumentioning

confidence: 99%

Section: Remark 1 (I) It Is Mentioned In Gong Wang and Yumentioning

confidence: 99%

See 2 more Smart Citations

Starlike mappings on bounded balanced domains with C¹-plurisubharmonic defining functions

Hamada¹

2000

Pacific J. Math.

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“…This fact is no longer true in higher dimensions (see [17]). Nevertheless, Barnard, FitzGerald and Gong showed that a similar assertion can be established for star-like mappings on the unit ball of a Banach space, normalized at the origin (see [12]). An improved result was obtained in [19] for the so-called strongly star-like mappings.…”

Section: A Parametric Representation Of Semicomplete Vector Fields Wimentioning

confidence: 69%

Complex dynamical systems and the geometry of domains in Banach spaces

Elin

Reich

Shoikhet

2004

Dissertationes Math.

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consisting of all the holomorphic self-mappings of D, is a semigroup with respect to composition. Definition 1.1.2. A mapping f ∈ Hol(D, Ω) is said to be univalent on D if for each pair of distinct points x 1 and x 2 in D we have f (x 1 ) = f (x 2 ).In this case one can define the inverse mappingIt is well known (see, for example, [39]) that if X = C n and Y = C m are finite dimensional complex spaces, then f :However, this fact is no longer true in the infinite dimensional case (see counterexamples in [79,4]). Therefore, in the general case a univalent mappingIt is also known (see, for example, [29,49]is a linear isomorphism between X and Y . In this situation we will say that D and Ω are biholomorphically equivalent.As we have already mentioned, in this case X and Y must be linearly isomorphic Banach spaces. Generally speaking, the converse is not true. That is, even if X and Y are isomorphic Banach spaces and f ∈ Hol(D, Y ) has at each point x in D a continuously invertible Fréchet derivative, the mapping f need not be univalent on D.Nevertheless, in this case the mapping f is biholomorphic on a neighborhood of each x ∈ D by the inverse function theorem. In this situation we will say that f ∈ Hol(D, Ω) is locally biholomorphic.The set of all univalent mappings from a domain D ⊂ X into X will be denoted by Univ(D). For the special case when D is the open unit ball of X, the subset of Univ(D) normalized by the conditions f (0) = 0 and f (0) = I will be denoted by S(D). This notation conforms to the one used in the classical onedimensional case, when D = ∆ = {z ∈ C : |z| < 1}. In this case we simply writeThat is, S consists of all the mappings f ∈ Univ(∆) such that f has the following Taylor series at the origin: Convex, star-shaped and spiral-shaped domainsDefinition 1.2.1. A set M in X is called convex if for each pair of points w 1 and w 2 in M , the line segment joining w 1 and w 2 is contained in M , i.e., for each t ∈ [0, 1], the point w = (1 − t)w 1 + tw 2 is also in M .If D is a domain in X, then f ∈ Univ(D) is said to be a convex mapping on D if its image f (D) = Ω is a convex domain in X.Geometry of domains in Banach spaces 9 Definition 1.2.2. A set M in X is called star-shaped (with respect to the origin) if given any w ∈ M , the point w t = tw also belongs to M for every t with 0 ≤ t ≤ 1. That is, if M contains w, then it also contains the entire line segment joining w to the origin. In this definition the origin is in Ω. If, in particular, the origin belongs to Ω, then we will say that f is star-like, to make our definition consistent with the classical one. In this case the mapping f has a null point τ in D. The set of all biholomorphic mappings on D which are star-shaped on D will be denoted by Star(D).If, in addition, there is τ ∈ D such that f (τ ) = 0, (1.2.1) then we will write f ∈ S * τ (D). Of course, in this case such a point τ is unique because. Again, in the one-dimensional case, when X = C, we will simply write S * to denote the family of all biholomorphic (univalent) star-like functions...

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Distortion of Biholomorphic Maps on Homogeneous Domains inJ*-Algebras

Włodarczyk

Szałowska

1997

Journal of Mathematical Analysis and Applications

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The growth and 1∕4-theorems for starlike mappings in Cⁿ

Cited by 39 publications

References 2 publications

Starlike mappings on bounded balanced domains with C¹-plurisubharmonic defining functions

Starlike mappings on bounded balanced domains with C¹-plurisubharmonic defining functions

Complex dynamical systems and the geometry of domains in Banach spaces

Distortion of Biholomorphic Maps on Homogeneous Domains inJ*-Algebras

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The growth and 1∕4-theorems for starlike mappings in Cn

Cited by 39 publications

References 2 publications

Starlike mappings on bounded balanced domains with C1-plurisubharmonic defining functions

Starlike mappings on bounded balanced domains with C1-plurisubharmonic defining functions

Complex dynamical systems and the geometry of domains in Banach spaces

Distortion of Biholomorphic Maps on Homogeneous Domains inJ*-Algebras

Contact Info

Product

Resources

About

The growth and 1∕4-theorems for starlike mappings in Cⁿ

Starlike mappings on bounded balanced domains with C¹-plurisubharmonic defining functions

Starlike mappings on bounded balanced domains with C¹-plurisubharmonic defining functions