scholarly journals The Nevanlinna–Pick problem on the closed unit disk: Minimal norm rational solutions of low degree

2012 ◽  
Vol 236 (13) ◽  
pp. 3123-3136 ◽  
Author(s):  
Vladimir Bolotnikov ◽  
Stephen P. Cameron
Keyword(s):  
Unit Disk ◽  
Rational Solutions ◽  
Minimal Norm ◽  
Closed Unit Disk ◽  
Low Degree ◽  
Pick Problem

scholarly journals Inner composition of analytic mappings on the unit disk

1991 ◽  
Vol 14 (2) ◽  
pp. 221-226 ◽  
Author(s):  
John Gill
Keyword(s):  
Fixed Point ◽  
Unit Disk ◽  
Analytic Functions ◽  
Continued Fraction ◽  
Unique Fixed Point ◽  
Iteration Theory ◽  
Basic Theorem ◽  
Closed Unit Disk ◽  
Compositional Structure ◽  
Compositional Structures

A basic theorem of iteration theory (Henrici [6]) states thatfanalytic on the interior of the closed unit diskDand continuous onDwithInt(D)f(D)carries any pointz ϵ Dto the unique fixed pointα ϵ Doff. That is to say,fn(z)→αasn→∞. In [3] and [5] the author generalized this result in the following way: LetFn(z):=f1∘…∘fn(z). Thenfn→funiformly onDimpliesFn(z)λ, a constant, for allz ϵ D. This kind of compositional structure is a generalization of a limit periodic continued fraction. This paper focuses on the convergence behavior of more general inner compositional structuresf1∘…∘fn(z)where thefj's are analytic onInt(D)and continuous onDwithInt(D)fj(D), but essentially random. Applications include analytic functions defined by this process.

scholarly journals Strong Differential Subordination Results for Multivalent Analytic Functions Associated with Dziok-Srivastava Operator

2019 ◽  
pp. 159-168
Author(s):  
Abbas Kareem Wanas ◽  
Hala Abbas Mehdi
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Unit Disk ◽  
Strong Differential Subordination

In this paper, by making use of the principle of strong subordination, we establish some interesting properties of multivalent analytic functions defined in the open unit disk and closed unit disk of the complex plane associated with Dziok-Srivastava operator.

scholarly journals New Applications of the Bernardi Integral Operator

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1180
Author(s):  
Shigeyoshi Owa ◽  
H. Özlem Güney
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Closed Unit Disk ◽  
New Applications

Let A ( p , n ) be the class of f ( z ) which are analytic p-valent functions in the closed unit disk U ¯ = z ∈ C : z ≤ 1 . The expression B − m − λ f ( z ) is defined by using fractional integrals of order λ for f ( z ) ∈ A ( p , n ) . When m = 1 and λ = 0 , B − 1 f ( z ) becomes Bernardi integral operator. Using the fractional integral B − m − λ f ( z ) , the subclass T p , n α s , β , ρ ; m , λ of A ( p , n ) is introduced. In the present paper, we discuss some interesting properties for f ( z ) concerning with the class T p , n α s , β , ρ ; m , λ . Also, some interesting examples for our results will be considered.

On a Problem Related to The Conjecture of Sendov About The Critical Points of a Polynomial

1987 ◽  
Vol 30 (4) ◽  
pp. 476-480
Author(s):  
Q. I. Rahman ◽  
Q. M. Tariq
Keyword(s):  
Unit Disk ◽  
Critical Points ◽  
Closed Unit Disk

AbstractLet P be a polynomial of degree n having all its zeros in the closed unit disk. Given that a is a zero (of P) of multiplicity k we seek to determine the radius ρ(n; k; a) of the smallest disk centred at a containing at least k zeros of the derivative P'. In the case k = 1 the answer has been conjectured to be 1 and is known to be true for n ≦ 5. We prove that ρ(n; k; a) ≦ 2k/(k + 1) for arbitrary k ∊ N and n ≦ k + 4.

scholarly journals THE MAPPING CLASS GROUP OF A DISK WITH INFINITELY MANY HOLES

2006 ◽  
Vol 15 (01) ◽  
pp. 21-29 ◽  
Author(s):  
PAUL FABEL
Keyword(s):  
Topological Group ◽  
Unit Disk ◽  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Closed Unit Disk ◽  
Completely Metrizable ◽  
Metrizable Topological Group

A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy.

Univalent Harmonic Ring Mappings Vanishing on the Interior Boundary

1992 ◽  
Vol 44 (2) ◽  
pp. 308-323 ◽  
Author(s):  
Walter Hengartner ◽  
Jan Szynal
Keyword(s):  
Unit Disk ◽  
Harmonic Mappings ◽  
Closed Unit Disk ◽  
Interior Boundary

AbstractWe give a characterization of univalent positively oriented harmonic mappings ƒ defined on an exterior neighbourhood of the closed unit disk { z: | z| ≤1} such that .

A Tauberian Theorem for the General Euler-Borel Summability Method

1992 ◽  
Vol 44 (5) ◽  
pp. 1100-1120 ◽  
Author(s):  
Laying Tam
Keyword(s):  
Positive Integer ◽  
Unit Disk ◽  
Tauberian Theorem ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Borel Summability ◽  
Closed Unit Disk ◽  
Tauberian Condition ◽  
Image Position ◽  
General Conditions

AbstractOur main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m > n, (m— n)n-p(f)→0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.

scholarly journals GEOMETRIC LIMITS OF MANDELBROT AND JULIA SETS UNDER DEGREE GROWTH

2012 ◽  
Vol 22 (12) ◽  
pp. 1250301 ◽  
Author(s):  
SUZANNE HRUSKA BOYD ◽  
MICHAEL J. SCHULZ
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Julia Sets ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Geometric Limit ◽  
The Family ◽  
Mandelbrot Sets

First, for the family Pn,c(z) = zn + c, we show that the geometric limit of the Mandelbrot sets Mn(P) as n → ∞ exists and is the closed unit disk, and that the geometric limit of the Julia sets J(Pn,c) as n tends to infinity is the unit circle, at least when |c| ≠ 1. Then, we establish similar results for some generalizations of this family; namely, the maps z ↦ zt + c for real t ≥ 2 and the rational maps z ↦ zn + c + a/zn.

scholarly journals Holomorphic isometries between products of complex unit balls

2017 ◽  
Vol 28 (09) ◽  
pp. 1740010 ◽  
Author(s):  
Shan Tai Chan ◽  
Ming Xiao ◽  
Yuan Yuan
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Closed Unit Disk ◽  
Holomorphic Map ◽  
Proper Holomorphic Map ◽  
Poincaré Disk

We first give an exposition on holomorphic isometries from the Poincaré disk to polydisks and from the Poincaré disk to the product of the Poincaré disk with a complex unit ball. As an application, we provide an example of proper holomorphic map from the unit disk to the complex unit ball that is irrational, algebraic and holomorphic on a neighborhood of the closed unit disk. We also include some new results on holomorphic isometries.

scholarly journals Effectiveness of Basic Sets of Goncarov and Related Polynomials

2021 ◽  
Author(s):  
Jerome A. Adepoju
Keyword(s):  
Unit Disk ◽  
Binomial Coefficients ◽  
Derivative Operator ◽  
Closed Unit Disk ◽  
Simple Sets ◽  
Sets Theory ◽  
Basic Sets

The Chapter presents diverse but related results to the theory of the proper and generalized Goncarov polynomials. Couched in the language of basic sets theory, we present effectiveness properties of these polynomials. The results include those relating to simple sets of polynomials whose zeros lie in the closed unit disk U=z:z≤1.. They settle the conjecture of Nassif on the exact value of the Whittaker constant. Results on the proper and generalized Goncarov polynomials which employ the q-analogue of the binomial coefficients and the generalized Goncarov polynomials belonging to the Dq- derivative operator are also given. Effectiveness results of the generalizations of these sets depend on whether q<1 or q>1. The application of these and related sets to the search for the exact value of the Whittaker constant is mentioned.

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