scholarly journals Note on the uniqueness holomorphic function on the unit disk

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2321-2325 ◽  
Author(s):  
Tuğba Akyel ◽  
Tahir Azeroğlu
Keyword(s):  
Holomorphic Function ◽  
Unit Disk ◽  
Blaschke Product ◽  
Local Behavior ◽  
Finite Blaschke Product ◽  
Boundary Points ◽  
Finite Set

Let f be an holomorphic function the unit disk to itself. We provide conditions on the local behavior of f along boundary near a finite set of the boundary points that requires f to be a finite Blaschke product.

scholarly journals Representation with majorant of the Schwarz lemma at the boundary

2017 ◽  
Vol 101 (115) ◽  
pp. 191-196
Author(s):  
Bülent Örnek ◽  
Tuğba Akyel
Keyword(s):  
Holomorphic Function ◽  
Blaschke Product ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Schwarz Lemma ◽  
Local Behavior ◽  
Finite Blaschke Product ◽  
Boundary Points ◽  
Finite Set ◽  
The Schwarz Lemma

Let f be a holomorphic function in the unit disc and |f(z)?1| < 1 for |z| < 1. We generalize the uniqueness portion of Schwarz?s lemma and provide sufficient conditions on the local behavior of f near a finite set of boundary points that needed for f to be a finite Blaschke product.

Approximation by Unimodular Functions

1971 ◽  
Vol 23 (2) ◽  
pp. 257-269 ◽  
Author(s):  
Stephen Fisher
Keyword(s):  
Blaschke Product ◽  
Uniform Approximation ◽  
Unit Disc ◽  
Open Unit ◽  
Blaschke Products ◽  
Open Unit Disc ◽  
Pointwise Approximation ◽  
Finite Blaschke Product ◽  
Image Position ◽  
Restricted Zeros

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

scholarly journals Boundary behaviour of holomorphic functions on the cardioid domain with some applications

2019 ◽  
Vol 38 (7) ◽  
pp. 203-218
Author(s):  
Shatha Sami Alhily ◽  
_ Deepmala
Keyword(s):  
General Solution ◽  
Holomorphic Function ◽  
Unit Disk ◽  
Laplace Equation ◽  
Holomorphic Functions ◽  
Research Paper ◽  
Boundary Behaviour ◽  
Polar Function

The objective of this research paper is to show how the Bennan'sconjecture  become a useful tool  to construct a holomorphic function on the cardioid domain, and on the boundary of unit disk. Moreover , we have addressed some applications on the existence of cusp on the boundary of arising from integrability of conformalmaps through one of the polar function in the general solution of Laplace equation.

scholarly journals Fatou’s Associates

2020 ◽  
Vol 6 (3-4) ◽  
pp. 459-493
Author(s):  
Vasiliki Evdoridou ◽  
Lasse Rempe ◽  
David J. Sixsmith
Keyword(s):  
Entire Function ◽  
Blaschke Product ◽  
Entire Functions ◽  
Julia Set ◽  
Singular Values ◽  
Strong Relationship ◽  
Inner Function ◽  
Transcendental Entire Function ◽  
Fatou Component ◽  
Finite Blaschke Product

AbstractSuppose that f is a transcendental entire function, $$V \subsetneq {\mathbb {C}}$$ V ⊊ C is a simply connected domain, and U is a connected component of $$f^{-1}(V)$$ f - 1 ( V ) . Using Riemann maps, we associate the map $$f :U \rightarrow V$$ f : U → V to an inner function $$g :{\mathbb {D}}\rightarrow {\mathbb {D}}$$ g : D → D . It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain.

scholarly journals Domain Identification for Inverse Problem via Conformal Mapping and Fixed Point Methods in Two Dimensions

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Fagueye Ndiaye
Keyword(s):  
Fixed Point ◽  
Conformal Mapping ◽  
Holomorphic Function ◽  
Unit Disk ◽  
Bounded Domain ◽  
Cauchy Data ◽  
Two Dimensions ◽  
Inverse Eigenvalue Problem ◽  
Unknown Boundary ◽  
Domain Identification

In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known part Γ0 and a homogeneous Dirichlet condition on an unknown part Γ0 of the boundary of a bounded domain, Ω⊂ℝN. We consider variations in the eigenvalues and propose a conformal mapping tool to reconstruct a part of the boundary curve of the two-dimensional bounded domain based on the Cauchy data of a holomorphic function that maps the unit disk onto the unknown domain. The boundary values of this holomorphic function are obtained by solving a nonlocal differential Bessel equation. Then, the unknown boundary is obtained as the image of the boundary of the unit disk by solving an ill-posed Cauchy problem for holomorphic functions via a regularized power expansion. The Cauchy data were restricted to a nonvanishing function and to the normal derivatives without zeros. We prove the existence and uniqueness of the holomorphic function being considered and use the fixed-point method to numerically analyze the results of convergence. We’ll calculate the eigenvalues and compare the result with the shape obtained via minimization functional method, as developed in a previous study. Further, we’ll observe via simulations the shape of Γ and if it preserves its properties with varying the eigenvalues.

Sign in / Sign up

Export Citation Format

Share Document