Behavior of Coefficients of Inverses of α-Spiral Functions

1986 ◽  
Vol 38 (6) ◽  
pp. 1329-1337 ◽  
Author(s):  
Richard J. Libera ◽  
Eligiusz J. Złotkiewicz
Keyword(s):  
Unit Disk ◽  
Series Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Maclaurin Series ◽  
Koebe Function ◽  
One To One ◽  
Image Position

If f(z) is univalent (regular and one-to-one) in the open unit disk Δ, Δ = {z ∊ C:│z│ < 1}, and has a Maclaurin series expansion of the form(1.1)then, as de Branges has shown, │ak│ = k, for k = 2, 3, … and the Koebe function.(1.1)serves to show that these bounds are the best ones possible (see [3]). The functions defined above are generally said to constitute the class .

scholarly journals Disk-like functions

1970 ◽  
Vol 11 (2) ◽  
pp. 251-256
Author(s):  
Richard J. Libera
Keyword(s):  
Unit Disk ◽  
Geometric Property ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Image Position

The class s of functions f(z) which are regular and univalent in the open unit disk △ = {z: |z| < 1} each normalized by the conditionshas been studied intensively for over fifty years. A large and very successful portion of this work has dealt with subclasses of L characterized by some geometric property of f[Δ], the image of Δ under f(z), which is expressible in analytic terms. The class of starlike functions in L is one of these [3]; f(z) is starlike with respect to the origin if the segment [0,f(z)] is in f[Δ] for every z in Δ and this condition is equivalent to requiring that have a positive real part in Δ.

Almost-Bounded Holomorphic Functions with Prescribed Ambiguous Points

1964 ◽  
Vol 16 ◽  
pp. 231-240 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Image Position ◽  
Bounded Holomorphic ◽  
Jordan Arcs ◽  
Ambiguous Point

Let f be a function mapping the open unit disk D into the extended complex plane. A point ζ on the unit circle C is called an ambiguous point of f if there exist two Jordan arcs J1 and J2, each having an endpoint at ζ and lying, except for ζ, in D, such that

scholarly journals Multipliers of Bergman spaces into Lebesgue spaces

1986 ◽  
Vol 29 (1) ◽  
pp. 125-131 ◽  
Author(s):  
Daniel H. Luecking
Keyword(s):  
Lebesgue Measure ◽  
Unit Disk ◽  
Analytic Functions ◽  
Lebesgue Space ◽  
Borel Measure ◽  
Bergman Spaces ◽  
Open Unit ◽  
Open Unit Disk ◽  
Lebesgue Spaces ◽  
Image Position

Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0<p<+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such thatprovided 0<p≦q.

Zero Tracts of Blaschke Products

1966 ◽  
Vol 18 ◽  
pp. 1072-1078 ◽  
Author(s):  
C. N. Linden ◽  
H. Somadasa
Keyword(s):  
Unit Disk ◽  
Blaschke Product ◽  
Open Unit ◽  
Complex Numbers ◽  
Open Unit Disk ◽  
Blaschke Products ◽  
Image Position

Let ﹛an﹜ be a sequence of complex numbers such thatandThen {an} is called a Blaschke sequence. For each Blaschke sequence {an} a Blaschke product is defined asThus a Blaschke product B(z, ﹛an﹜) is a function regular in the open unit disk D = {z: |z| < 1﹜ and having a zero at each point of the sequence ﹛an﹜.

On Algebras Generated by Composition Operators

1974 ◽  
Vol 26 (5) ◽  
pp. 1234-1241 ◽  
Author(s):  
J. A. Cima ◽  
W. R. Wogen
Keyword(s):  
Banach Spaces ◽  
Unit Disk ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
Group Of Automorphisms ◽  
Image Position

Let Δ be the open unit disk in the complex plane and let be the group of automorphisms of Δ onto Δ, define byThe Banach spaces Hp = Hp(Δ), 1 ≦ p < ∞, are the Hardy spaces of functions analytic in Δ with their integral p means bounded,

Green's Potentials with Prescribed Boundary Values

1977 ◽  
Vol 29 (4) ◽  
pp. 681-686
Author(s):  
Jang-Mei G. Wu
Keyword(s):  
Green’S Function ◽  
Unit Disk ◽  
Mass Distribution ◽  
Green's Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Boundary Values ◽  
Image Position

Let U, C denote the open unit disk and unit circumference, respectively and G(z, w) be the Green's function on U. We say v is the Green's potential of a mass distribution v on U if

Normal Functions and Non-Tangential Boundary Arcs

1966 ◽  
Vol 18 ◽  
pp. 256-264 ◽  
Author(s):  
P. Lappan ◽  
D. C. Rung
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Hyperbolic Geometry ◽  
Open Unit ◽  
Open Unit Disk ◽  
Normal Functions ◽  
Image Position

Let D and C denote respectively the open unit disk and the unit circle in the complex plane. Further, γ = z(t), 0 ⩽ t ⩽ 1, will denote a simple continuous arc lying in D except for Ƭ = z(l) ∈ C, and we shall say that γ is a boundary arc at Ƭ.We use extensively the notions of non-Euclidean hyperbolic geometry in D and employ the usual metricwhere a and b are elements of D. For a ∈ D and r > 0 letFor details we refer the reader to (4).

Univalent α-Spiral Functions

1967 ◽  
Vol 19 ◽  
pp. 449-456 ◽  
Author(s):  
Richard J. Libera
Keyword(s):  
Unit Disk ◽  
Simple Zero ◽  
Open Unit ◽  
Open Unit Disk ◽  
Image Position

Suppose ƒ is regular in the open unit disk Δ, |z| < 1, and has a simple zero at the origin and no other zeros. Špaček (15) essentially showed that ƒ is univalent in Δ if and only ifsuch that 0 < r < 1 and 0 < t2 — t1 ⩽ 2π.

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

scholarly journals Coefficients of a Comprehensive Subclass of Meromorphic Bi-Univalent Functions Associated with the Faber Polynomial Expansion

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 27
Author(s):  
Hari Mohan Srivastava ◽  
Ahmad Motamednezhad ◽  
Safa Salehian
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Expansion Method ◽  
Upper Bounds ◽  
Polynomial Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Faber Polynomial ◽  
The Subject ◽  
Polynomial Expansion Method

In this paper, we introduce a new comprehensive subclass ΣB(λ,μ,β) of meromorphic bi-univalent functions in the open unit disk U. We also find the upper bounds for the initial Taylor-Maclaurin coefficients |b0|, |b1| and |b2| for functions in this comprehensive subclass. Moreover, we obtain estimates for the general coefficients |bn|(n≧1) for functions in the subclass ΣB(λ,μ,β) by making use of the Faber polynomial expansion method. The results presented in this paper would generalize and improve several recent works on the subject.

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