Functions which Map the Interior of the Unit Circle Upon Simple Regions

Alexander, James W.

doi:10.2307/2007212

Cited by 355 publications

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“…Star-like functions on the unit disk. The concept of univalent star-like functions was first introduced by Alexander [8] in 1915. In 1921 Nevanlinna [58] made a more detailed study of this class.…”

Section: Historical Sketchmentioning

confidence: 99%

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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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Section: Historical Sketchmentioning

confidence: 99%

“…Here we quote another classical result due to Alexander [8] which provides an analytic connection between convex and star-like functions. As we mentioned above, in the study of the functions of the form…”

Section: Convex and Close-to-convex Functionsmentioning

confidence: 99%

Complex dynamical systems and the geometry of domains in Banach spaces

Elin

Reich

Shoikhet

2004

Dissertationes Math.

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“…The operators F 1 α,β (z) and F 1 α,α (z) were studied by Breaz and Breaz (see [4]) and Pescar (see [12]), respectively. Upon setting β = 1 and α = β = 1 in F 1 α,β (z), we can obtain the operators F 1 α,1 (z) and F 1 1,1 (z) which were studied by Breaz and Breaz (see [3]) and Alexander (see [1]), respectively. Furthermore, in their special cases when p = n = β = 1, and 1/α instead of α i = α for all i = 1, n, the p−valent integral operator in (7) would obviously reduce to the operator F 1 1/α,1 (z) which was studied Pescar and Owa (see [13]), for α ∈ [0, 1] special case of the operator F 1 1/α,1 (z) was studied by Miller, Mocanu and Reade (see [11]).…”

Section: Introductionmentioning

confidence: 99%

The Relationships Between p−valent Functions and Univalent Functions

Çağlar

2015

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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“…It is an important observation due to Alexander [2] that f (z) is convex if and only if g(z) = zf ′ (z) is starlike. The mapping g → f is sometimes called the Alexander transformation and will be denoted by J 1 [f ] in the sequel.…”

Section: Introductionmentioning

confidence: 99%

On power deformations of univalent functions

Kim

Sugawa

2011

Monatsh Math

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Abstract. For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f ′ (0)−1 = 0 and f (z) = 0, 0 < |z| < 1, we consider the power deformation f c (z) = z(f (z)/z) c for a complex number c. We determine those values c for which the operator f → f c maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf ′ (z)/f (z), |z| < 1, for the class in most cases which we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.

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Functions which Map the Interior of the Unit Circle Upon Simple Regions

Cited by 355 publications

References 0 publications

Complex dynamical systems and the geometry of domains in Banach spaces

Complex dynamical systems and the geometry of domains in Banach spaces

The Relationships Between p−valent Functions and Univalent Functions

On power deformations of univalent functions

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