scholarly journals On Coefficient Functionals for Functions with Coefficients Bounded by 1

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 491
Author(s):  
Paweł Zaprawa ◽  
Anna Futa ◽  
Magdalena Jastrzębska
Keyword(s):  
Unit Disk ◽  
Upper Bound ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Extremal Functions ◽  
Hankel Determinant ◽  
Schwarz Function ◽  
Positive Real ◽  
Special Case

In this paper, we discuss two well-known coefficient functionals a 2 a 4 - a 3 2 and a 4 - a 2 a 3 . The first one is called the Hankel determinant of order 2. The second one is a special case of Zalcman functional. We consider them for functions in the class Q R ( 1 2 ) of analytic functions with real coefficients which satisfy the condition ( ) f ( z ) z > 1 2 for z in the unit disk Δ . It is known that all coefficients of f ∈ Q R ( 1 2 ) are bounded by 1. We find the upper bound of a 2 a 4 - a 3 2 and the bound of | a 4 - a 2 a 3 | . We also consider a few subclasses of Q R ( 1 2 ) and we estimate the above mentioned functionals. In our research two different methods are applied. The first method connects the coefficients of a function in a given class with coefficients of a corresponding Schwarz function or a function with positive real part. The second method is based on the theorem of formulated by Szapiel. According to this theorem, we can point out the extremal functions in this problem, that is, functions for which equalities in the estimates hold. The obtained estimates significantly extend the results previously established for the discussed classes. They allow to compare the behavior of the coefficient functionals considered in the case of real coefficients and arbitrary coefficients.

Second hankel determinat for certain analytic functions satisfying subordinate condition

2018 ◽  
Vol 68 (2) ◽  
pp. 463-471
Author(s):  
Erhan Deniz ◽  
Levent Budak
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Real Axis ◽  
Upper Bound ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Hankel Determinant ◽  
Positive Real ◽  
Toeplitz Determinants ◽  
Second Hankel Determinant

Abstract In this paper, we introduce and investigate the following subclass $$\begin{array}{} \displaystyle 1+\frac{1}{\gamma }\left( \frac{zf'(z)+\lambda z^{2}f''(z)}{\lambda zf'(z)+(1-\lambda )f(z)}-1\right) \prec \varphi (z)\qquad\left( 0\leq \lambda \leq 1,\gamma \in \mathbb{C} \smallsetminus \{0\}\right) \end{array} $$ of analytic functions, φ is an analytic function with positive real part in the unit disk 𝔻, satisfying φ (0) = 1, φ '(0) > 0, and φ (𝔻) is symmetric with respect to the real axis. We obtain the upper bound of the second Hankel determinant | a2a4− $\begin{array}{} a^2_3 \end{array} $ | for functions belonging to the this class is studied using Toeplitz determinants. The results, which are presented in this paper, would generalize those in related works of several earlier authors.

scholarly journals Coefficient estimates of certain subclasses of analytic functions associated with Hohlov operator

2019 ◽  
pp. 2150021
Author(s):  
P. Gochhayat ◽  
A. Prajapati ◽  
A. K. Sahoo
Keyword(s):  
Upper Bound ◽  
Analytic Functions ◽  
Gauss Hypergeometric Function ◽  
Open Unit ◽  
Sufficient Condition ◽  
Extremal Functions ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Coefficient Estimates ◽  
Extremal Value

A typical quandary in geometric functions theory is to study a functional composed of amalgamations of the coefficients of the pristine function. Conventionally, there is a parameter over which the extremal value of the functional is needed. The present paper deals with consequential functional of this type. By making use of Hohlov operator, a new subclass [Formula: see text] of analytic functions defined in the open unit disk is introduced. For both real and complex parameter, the sharp bounds for the Fekete–Szegö problems are found. An attempt has also been taken to found the sharp upper bound to the second and third Hankel determinant for functions belonging to this class. All the extremal functions are express in term of Gauss hypergeometric function and convolution. Finally, the sufficient condition for functions to be in [Formula: see text] is derived. Relevant connections of the new results with well-known ones are pointed out.

scholarly journals Some characteristic properties of analytic functions

2016 ◽  
Vol 24 (1) ◽  
pp. 353-369
Author(s):  
R. K. Raina ◽  
Poonam Sharma ◽  
G. S. Sălăgean
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Integral Representations ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Bilinear Mapping ◽  
Initial Coefficient

AbstractIn this paper, we consider a class L(λ, μ; ϕ) of analytic functions f defined in the open unit disk U satisfying the subordination condition that,where is the Sălăgean operator and ϕ(z) is a convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius and subordination results, and an inclusion result for the above class when the function ϕ(z) is a bilinear mapping in the open unit disk. For these functions f (z) ; sharp bounds for the initial coefficient and for the Fekete-Szegö functional are determined, and also some integral representations are given.

scholarly journals Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Hameed Ur Rehman ◽  
Maslina Darus ◽  
Jamal Salah
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Positive Real Part ◽  
Polynomial Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Faber Polynomial ◽  
Coefficient Problems

In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. This formula is applied to a certain class of bi-univalent functions and solve the n -th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its n -th term.

scholarly journals Third-Order Hankel Determinant for Certain Class of Analytic Functions Related with Exponential Function

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 501 ◽  
Author(s):  
Hai-Yan Zhang ◽  
Huo Tang ◽  
Xiao-Meng Niu
Keyword(s):  
Unit Disk ◽  
Exponential Function ◽  
Upper Bound ◽  
Analytic Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Third Order ◽  
The Third

Let S l * denote the class of analytic functions f in the open unit disk D = { z : | z | < 1 } normalized by f ( 0 ) = f ′ ( 0 ) − 1 = 0 , which is subordinate to exponential function, z f ′ ( z ) f ( z ) ≺ e z ( z ∈ D ) . In this paper, we aim to investigate the third-order Hankel determinant H 3 ( 1 ) for this function class S l * associated with exponential function and obtain the upper bound of the determinant H 3 ( 1 ) . Meanwhile, we give two examples to illustrate the results obtained.

scholarly journals A note on neighborhoods of analytic functions having positive real part

1990 ◽  
Vol 13 (3) ◽  
pp. 425-429 ◽  
Author(s):  
Janice B. Walker
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Class Of Functions

LetPdenote the set of all functions analytic in the unit diskD={z||z|<1}having the formp(z)=1+∑k=1∞pkzkwithRe{p(z)}>0. Forδ≥0, letNδ(p)be those functionsq(z)=1+∑k=1∞qkzkanalytic inDwith∑k=1∞|pk−qk|≤δ. We denote byP′the class of functions analytic inDhaving the formp(z)=1+∑k=1∞pkzkwithRe{[zp(z)]′}>0. We show thatP′is a subclass ofPand detemineδso thatNδ(p)⊂Pforp∈P′.

scholarly journals On a Subclass of Analytic Functions Related to a Hyperbola

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Jagannath Patel ◽  
Ashok Kumar Sahoo
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Upper Bound ◽  
Analytic Functions ◽  
Sufficient Condition ◽  
Hankel Determinant ◽  
New Class ◽  
Second Hankel Determinant

The object of the present investigation is to solve Fekete-Szegö problem and determine the sharp upper bound to the second Hankel determinant for a new classℛ̃(a,c,ρ)of analytic functions in the unit disk. We also obtain a sufficient condition for an analytic function to be in this class.

scholarly journals On Certain Subclasses of Analytic Functions Involving Carlson-Shaffer Operator and Related to Lemniscate of Bernoulli

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jagannath Patel ◽  
Ashok Kumar Sahoo
Keyword(s):  
Unit Disk ◽  
Upper Bound ◽  
Analytic Functions ◽  
Sufficient Condition ◽  
Hankel Determinant ◽  
Lemniscate Of Bernoulli ◽  
New Class ◽  
Second Hankel Determinant

The object of the present investigation is to solve the Fekete-Szegö problem and determine the sharp upper bound to the second Hankel determinant for a new class R(a,c) of analytic functions involving the Carlson-Shaffer operator in the unit disk. We also obtain a sufficient condition for normalized analytic functions in the unit disk to be in this class.

An Inequality Concerning Analytic Functions with a Positive Real Part

1969 ◽  
Vol 21 ◽  
pp. 1172-1177 ◽  
Author(s):  
Thomas H. MacGregor
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Image Position

This paper contains an inequality about functions which are analytic and have a positive real part in the unit disk. A first consequence of the inequality is the fact that if is analytic for |z| < 1 and has values lying in a strip of width δ. This result is known and was first proved by Tammi (1).Our second theorem is a generalization of this. Namely, ifis analytic for |z| < 1 and satisfies Re{zmf(m>(z)} ≧ A andthenconverges.Another application of our fundamental inequality is the following. Let be analytic for |z| < 1 and satisfy Re p(z) ≧ 0 and set and .

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