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 On the Boundary Behavior of Holomorphic Functions in the Unit Disk

1962 ◽  
Vol 20 ◽  
pp. 95-103 ◽  
Author(s):  
Claude Marie Faust
Keyword(s):  
Holomorphic Function ◽  
Unit Disk ◽  
Boundary Behavior ◽  
Holomorphic Functions ◽  
Hyperbolic Distance ◽  
Infinitely Many Solutions

Let f(z) be a holomorphic function defined in the unit disk |z|<1, which we shall denote by D. Let Σ be a subset of D, whose closure has at least one point in common with C, the circumference of the unit disk. The set of all values a such that the equation f(z) = a has infinitely many solutions in Σ is called the range of f(z) in Σ, and is denoted by R(f, Σ). Let τ be a point of C, and let {zn ) be a sequence of points in D with the properties: . The non-Euclidean (hyperbolic) distance ρ(zn, zn+1 ) between two points zn and z n+1 of the sequence is defined to be equal to (cf.[3], Ch. II).

scholarly journals Uniformly perfect sets and distortion of holomorphic functions

2001 ◽  
Vol 164 ◽  
pp. 17-33 ◽  
Author(s):  
Jian-Hua Zheng
Keyword(s):  
Holomorphic Function ◽  
Unit Disk ◽  
Boundary Point ◽  
Univalent Functions ◽  
Holomorphic Functions ◽  
Universal Covering ◽  
Open Set ◽  
Uniformly Perfect ◽  
Hyperbolic Domain ◽  
Uniform Perfectness

We investigate the uniform perfectness on a boundary point of a hyperbolic open set and distortion of a holomorphic function from the unit disk Δ into a hyperbolic domain with a uniformly perfect boundary point, especially of a universal covering map of such a domain from Δ, and we obtain similar results to celebrated Koebe’s Theorems on univalent functions.

scholarly journals Stability of a generalized n-variable mixed-type functional equation in fuzzy modular spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Murali Ramdoss ◽  
Divyakumari Pachaiyappan ◽  
Choonkil Park ◽  
Jung Rye Lee
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Functional Equations ◽  
Mixed Type ◽  
Research Paper ◽  
Fixed Point Method ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Modular Spaces

AbstractThis research paper deals with general solution and the Hyers–Ulam stability of a new generalized n-variable mixed type of additive and quadratic functional equations in fuzzy modular spaces by using the fixed point method.

scholarly journals Coefficients problems for families of holomorphic functions related to hyperbola

2020 ◽  
Vol 70 (3) ◽  
pp. 605-616
Author(s):  
Stanisława Kanas ◽  
Vali Soltani Masih ◽  
Ali Ebadian
Keyword(s):  
Unit Disk ◽  
Holomorphic Functions ◽  
Hankel Determinant ◽  
Coefficient Problem

AbstractWe consider a family of analytic and normalized functions that are related to the domains ℍ(s), with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation $\begin{array}{} \frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0 \lt s\le 1,\, |\varphi| \lt (\pi s)/2). \end{array}$ We mainly study a coefficient problem of the families of functions for which zf′/f or 1 + zf″/f′ map the unit disk onto a subset of ℍ(s) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

scholarly journals Function-Theoretic Metrics and Boundary Behaviour of Functions Meromorphic or Holomorphic in the Unit Disk

1972 ◽  
Vol 45 ◽  
pp. 109-117
Author(s):  
Shinji Yamashita
Keyword(s):  
Unit Disk ◽  
Boundary Behaviour ◽  
Arc Length

The metrics to which the title of the present paper refers are expressed in the form of elements of arc length as follows:

scholarly journals ON THE ANNIHILATOR OF A DOLBEAULT GROUP

2010 ◽  
Vol 21 (03) ◽  
pp. 317-331
Author(s):  
IMRE PATYI
Keyword(s):  
Holomorphic Function ◽  
Open Subset ◽  
Cohomology Group ◽  
Holomorphic Functions ◽  
Dolbeault Cohomology ◽  
Infinite Dimensional ◽  
Continuous Character

We show that any Dolbeault cohomology group Hp,q(D), p ≥ 0, q ≥ 1, of an open subset D of a closed finite codimensional complex Hilbert submanifold of ℓ2 is either zero or infinite dimensional. We also show that any continuous character of the algebra of holomorphic functions of a closed complex Hilbert submanifold M of ℓ2 is induced by its evaluation at a point of M. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of ℓ2 admits a nowhere critical holomorphic function.

A POLAR DECOMPOSITION FOR HOLOMORPHIC FUNCTIONS ON A STRIP

2001 ◽  
Vol 33 (3) ◽  
pp. 309-319 ◽  
Author(s):  
KONRAD SCHMÜDGEN
Keyword(s):  
Holomorphic Function ◽  
Real Axis ◽  
Polar Decomposition ◽  
Holomorphic Functions ◽  
Heisenberg Algebra ◽  
Boundary Values ◽  
Operator Representation ◽  
The Real ◽  
Almost Everywhere

Let f be a holomorphic function on the strip {z ∈ [Copf ] : −α < Im z < α}, where α > 0, belonging to the class [Hscr ](α,−α;ε) defined below. It is shown that there exist holomorphic functions w1 on {z ∈ [Copf ] : 0 < Im z < 2α} and w2 on {z ∈ [Copf ] : −2α < Im z < 2α}, such that w1 and w2 have boundary values of modulus one on the real axis, and satisfy the relationsw1(z)=f(z-αi)w2(z-2αi) and w2(z+2αi)=f(z+αi)w1(z)for 0 < Im z < 2α, where f(z) := f(z). This leads to a ‘polar decomposition’ f(z) = uf(z + αi)gf(z) of the function f(z), where uf (z + αi) and gf(z) are holomorphic functions for −α < Im z < α, such that [mid ]uf(x)[mid ] = 1 and gf(x) [ges ] 0 almost everywhere on the real axis. As a byproduct, an operator representation of a q-deformed Heisenberg algebra is developed.

scholarly journals Composition operators on some Möbius invariant Banach spaces

2000 ◽  
Vol 62 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Shamil Makhmutov ◽  
Maria Tjani
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Disk ◽  
Composition Operator ◽  
Composition Operators ◽  
Bloch Space ◽  
Holomorphic Functions ◽  
Mean Oscillation

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

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