scholarly journals Solving one extreme problem in a class of locally single-leaf functions

2021 ◽  
pp. 58-69
Author(s):  
Ольга Евгеньевна Баранова
Keyword(s):  
Holomorphic Functions ◽  
Central Place ◽  
Accurate Estimation ◽  
Coefficient Problem ◽  
Conformal Maps ◽  
Taylor Coefficients ◽  
Single Leaf ◽  
Extreme Problem ◽  
Structural Formulas ◽  
Locally Conformal

Центральное место в теории конформных отображений занимает решение экстремальных задач на классах однолистных отображений. В известных классах нормированных голоморфных функций $S$ и $C$ решение «проблемы коэффициентов» связано с получением точных оценок модулей тейлоровских коэффициентов элементов классов. Аналогичные задачи ставятся для классов локально однолистных отображений. В.Г.Шеретов ввел в рассмотрение классы локально конформных отображений, генерируемых с помощью интегральных структурных формул из элементов классов $S$ и $C$. В статье решена задача о точной оценке модуля тейлоровского коэффициента в этом классе. The central place in the theory of conformal maps is occupied by the solution of extreme problems on classes of single-leaf maps. In the known classes of normalized holomorphic functions S and C, the solution of the "coefficient problem" is associated with obtaining accurate estimates of the modules of the Taylor coefficients of class elements. Similar problems are posed for classes of locally single-leaf mappings. V.G.Sheretov introduced classes of locally conformal mappings generated using integral structural formulas from elements of classes S and C. The article solves the problem of an accurate estimation of the modulus of the Taylor coefficient in this class.

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 A note on a space Hp, a of holomorphic functions

1987 ◽  
Vol 35 (3) ◽  
pp. 471-479
Author(s):  
H. O. Kim ◽  
S. M. Kim ◽  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Weak Type ◽  
Holomorphic Functions ◽  
Embedding Theorem ◽  
Type Operator ◽  
Taylor Coefficients ◽  
Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

scholarly journals Best approximation of holomorphic functions from hardy space in terms of Taylor coefficients

Filomat ◽  
2019 ◽  
Vol 33 (5) ◽  
pp. 1417-1424
Author(s):  
F.G. Abdullayev ◽  
G.A. Abdullayev ◽  
V.V. Savchuk
Keyword(s):  
Hardy Space ◽  
Polynomial Approximation ◽  
Best Approximation ◽  
Holomorphic Functions ◽  
Best Polynomial Approximation ◽  
Taylor Coefficients

We describe the set of holomorphic functions from the Hardy space Hq, 1 ? q ? ?, for which the best polynomial approximation En(f)q is equal to |f (n)(0)|=n!.

scholarly journals On radial variation of holomorphic functions with lp Taylor coefficients

1990 ◽  
Vol 33 (3) ◽  
pp. 475-481
Author(s):  
D. J. Hallenbeck ◽  
K. Samotij
Keyword(s):  
Holomorphic Functions ◽  
Positive Function ◽  
Radial Variation ◽  
Taylor Coefficients ◽  
Image Position

Suppose is holomorphic in Δ = {z:|z|<l} and (an)∈lp where 1≦p≦2. We prove that for k=1,2,…, and almost every θ. This result is sharp in the following sense: Let p∈[1,2] and ε(r) be a positive function defined on [0,1] such that limr→1-ε(r)=0. Then there exists a function holomorphic in Δ with (an)∈lp such thatfor each k>1/p.

scholarly journals Coefficient Estimates for a Subclass of Starlike Functions

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1646
Author(s):  
Dorina Răducanu
Keyword(s):  
Univalent Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Coefficient Estimates ◽  
Coefficient Problem ◽  
Taylor Coefficients ◽  
The Difference ◽  
Theoretical Results

In this note, we consider a subclass H3/2(p) of starlike functions f with f″(0)=p for a prescribed p∈[0,2]. Usually, in the study of univalent functions, estimates on the Taylor coefficients, Fekete–Szegö functional or Hankel determinats are given. Another coefficient problem which has attracted considerable attention is to estimate the moduli of successive coefficients |an+1|−|an|. Recently, the related functional |an+1−an| for the initial successive coefficients has been investigated for several classes of univalent functions. We continue this study and for functions f(z)=z+∑n=2∞anzn∈H3/2(p), we investigate upper bounds of initial coefficients and the difference of moduli of successive coefficients |a3−a2| and |a4−a3|. Estimates of the functionals |a2a4−a32| and |a4−a2a3| are also derived. The obtained results expand the scope of the theoretical results related with the functional |an+1−an| for various subclasses of univalent functions.

scholarly journals The extreme points of a class of functions with positive real part

1991 ◽  
Vol 44 (2) ◽  
pp. 253-261
Author(s):  
N. Samaris
Keyword(s):  
Extreme Point ◽  
Unit Disc ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Greatest Common Divisor ◽  
Positive Real ◽  
Taylor Coefficients ◽  
Class Of Functions

Let P1 be the class of holomorphic functions on the unit disc U = {z: |z| < 1} for which f(0) = 1 and Re f > 0. Let also Pn be the corresponding class on the unit disc Un. The inequality |ak| ≤ 2 is known for the Taylor coefficients in the class P1. In this paper, it is generalised for the class Pn. If ρ = (ρ1, ρ2, …, ρn), with ρ1, ρ2, …, ρn nonegative integers whose greatest common divisor is equal to 1, we describe the form of the functions f ∈ Pn under the restriction |aρ| = 2. Under the same restriction, we give conditions for a function to be an extreme point of the class Pn.

scholarly journals The Radon Inversion Problem for Holomorphic Functions in the Unit Disc

2021 ◽  
Author(s):  
P. Kot ◽  
P. Pierzchała
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Holomorphic Functions ◽  
Positive Function ◽  
Inversion Problem ◽  
Technical Improvement ◽  
Taylor Coefficients ◽  
Convergence And Divergence ◽  
Strictly Positive Function

AbstractThis paper deals with the so-called Radon inversion problem formulated in the following way: Given a $$p>0$$ p > 0 and a strictly positive function H continuous on the unit circle $${\partial {\mathbb {D}}}$$ ∂ D , find a function f holomorphic in the unit disc $${\mathbb {D}}$$ D such that $$\int _0^1|f(zt)|^pdt=H(z)$$ ∫ 0 1 | f ( z t ) | p d t = H ( z ) for $$z \in {\partial {\mathbb {D}}}$$ z ∈ ∂ D . We prove solvability of the problem under consideration. For $$p=2$$ p = 2 , a technical improvement of the main result related to convergence and divergence of certain series of Taylor coefficients is obtained.

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