scholarly journals Continued Fractions and Conformal Mappings for Domains with Angel Points

2018 ◽  
Vol 7 (4.7) ◽  
pp. 409
Author(s):  
Pyotr N. Ivanshin ◽  
. .
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Continued Fraction ◽  
Root Function ◽  
Conformal Mappings ◽  
Thin Domains ◽  
Root Functions ◽  
Mapping Sequence ◽  
Complex Half Plane ◽  
The Right

Here we construct the conformal mappings with the help of the continued fraction approximations. We first show that the method of [19] works for conformal mappings of the unit disk onto domains with acute external angles at the boundary. We give certain illustrative examples of these constructions. Next we outline the problem with domains which boudary possesses acute internal angles. Then we construct the method of rational root approximation in the right complex half-plane. First we construct the square root approximation and consider approximative properties of the mapping sequence in Theorem 1. Then we turn to the general case, namely, the continued fraction approximation of the rational root function in the complex right half-plane. These approximations converge to the algebraic root functions , , , . This is proved in Theorem 2 of the aricle. Thus we prove convergence of this method and construct conformal approximate mappings of the unit disk onto domains with angles and thin domains. We estimate the convergence rate of the approximation sequences. Note that the closer the point is to zero or infinity and the lower is the ratio k/N the worse is the approximation. Also we give the examples that illustrate the conformal mapping construction.  

The Number of Zeros of a Polynomial in a Half-Plane

1960 ◽  
Vol 56 (2) ◽  
pp. 132-147 ◽  
Author(s):  
A. Talbot
Keyword(s):  
Half Plane ◽  
Continued Fraction ◽  
Complex Polynomial ◽  
Continued Fraction Expansion ◽  
Left Half ◽  
Number Of Zeros ◽  
Full Discussion ◽  
The Subject ◽  
The Right

The determination of the number of zeros of a complex polynomial in a half-plane, in particular in the upper and lower, or right and left, half-planes, has been the subject of numerous papers, and a full discussion, with many references, is given in Marden (l) and Wall (2), where the basis for the determination is a continued-fraction expansion, or H.C.F. algorithm, in terms of which the number of zeros in one of the half-planes can be written down at once. In addition, determinantal formulae for the relevant elements of the algorithm can be obtained, and these lead to determinantal criteria for the number of zeros, including that of Hurwitz (3) for the right and left half-planes.

scholarly journals Radius of starlikeness for some classes containing non-univalent functions

2021 ◽  
pp. 2250009
Author(s):  
Shalu Yadav ◽  
Kanika Sharma ◽  
V. Ravichandran
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Univalent Function ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Radius Of Starlikeness ◽  
The Past ◽  
The Right ◽  
Starlike Univalent Function

A starlike univalent function [Formula: see text] is characterized by [Formula: see text]; several subclasses of starlike functions were studied in the past by restricting [Formula: see text] to take values in a region [Formula: see text] on the right-half plane, or, equivalently, by requiring [Formula: see text] to be subordinate to the corresponding mapping of the unit disk [Formula: see text] to the region [Formula: see text]. The mappings [Formula: see text], [Formula: see text], defined by [Formula: see text] and [Formula: see text] map the unit disk [Formula: see text] to certain nice regions in the right-half plane. For normalized analytic functions [Formula: see text] with [Formula: see text] and [Formula: see text] are subordinate to the function [Formula: see text] for some analytic functions [Formula: see text] and [Formula: see text], we determine the sharp radius for them to belong to various subclasses of starlike functions.

Coefficient functionals and radius problems of certain starlike functions

2021 ◽  
pp. 2250089
Author(s):  
Adiba Naz ◽  
Sushil Kumar ◽  
V. Ravichandran
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
Radius Problems ◽  
Second Hankel Determinant ◽  
The Right ◽  
Inverse Functions

Ma–Minda class (of starlike functions) consists of normalized analytic functions [Formula: see text] defined on the unit disk for which the image of the function [Formula: see text] is contained in some starlike region lying in the right-half plane. In this paper, we obtain the best possible bounds on some initial coefficients for the inverse functions of Ma–Minda starlike functions. Further, the bounds on the Fekete–Szegö functional and the second Hankel determinant are computed for such functions. In addition, some sharp radius estimates are also determined.

Starlike functions associated with a lune

2017 ◽  
Vol 10 (04) ◽  
pp. 1750064 ◽  
Author(s):  
Shweta Gandhi ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Half Plane ◽  
Complex Plane ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Lemniscate Of Bernoulli ◽  
The Right

Several subclasses of starlike functions are associated with regions in the right half plane of the complex plane, like half-plane, disks, sectors, parabolas and lemniscate of Bernoulli. For a normalized analytic function [Formula: see text] defined on the open unit disk [Formula: see text] belonging to certain well-known classes of functions associated with the above regions, we investigate the radius [Formula: see text] such that, for the function [Formula: see text], [Formula: see text] lies in the lune defined by [Formula: see text] for all [Formula: see text].

scholarly journals The problem of normal oscillations of a viscous stratified fluid with an elastic membrane

2021 ◽  
Vol 31 (2) ◽  
pp. 311-330
Author(s):  
D.O. Tsvetkov
Keyword(s):  
Half Plane ◽  
Model Problem ◽  
Stratified Fluid ◽  
Elastic Membrane ◽  
Spatial Problem ◽  
Scalar Model ◽  
Oscillation Spectrum ◽  
Complex Half Plane ◽  
The Right ◽  
Location Zone

Normal oscillations of a viscous stratified fluid partially filling an arbitrary vessel and bounded above by an elastic horizontal membrane are studied. In this case, we consider a scalar model problem that reflects the main features of the vector spatial problem. The characteristic equation for the eigenvalues of the model problem is obtained, the structure of the spectrum and the asymptotics of the branches of the eigenvalues are studied. Assumptions are made about the structure of the oscillation spectrum of a viscous stratified fluid bounded by an elastic membrane for an arbitrary vessel. It is proved that the spectrum of the problem is discrete, located in the right complex half-plane symmetrically with respect to the real axis, and has a single limit point $+\infty$. Moreover, the spectrum is localized in a certain way in the right half-plane, the location zone depends on the dynamic viscosity of the fluid.

On Some Extremal Problems on Linearly Invariant Classes

2005 ◽  
Vol 10 (2) ◽  
pp. 161-170
Author(s):  
E. G. Kiriyatzkii ◽  
J. Kirjackis
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Extremal Problems ◽  
Invariant Class ◽  
Definition Of ◽  
The Right ◽  
Variational Formulas ◽  
Omega Operator

In present paper the definition of linearly invariant class of analytical in the right half-plane is given and some extremal problems on introduced class are solved. For solving we use method based on variational formulas with specially introduced omega-operator, defined on these classes. In case when domain is unit disk similar linearly invariant classes were considered by Ch. Pommerenke, V. Starkov, E.G. Kiriyatzkii.

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.

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