Estimates for the Koebe Constant and the Second Coefficient for Some Classes of Univalent Functions

1980 ◽  
Vol 32 (6) ◽  
pp. 1311-1324 ◽  
Author(s):  
D. Bshouty ◽  
W. Hengartner ◽  
G. Schober
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
The Other ◽  
Open Unit ◽  
Open Unit Disk ◽  
Identity Mapping ◽  
The Best Possible Constant

Let S be the set of all normalized univalent analytic functions ƒ(z) = z + a2z2 + … in the open unit disk U. Then ƒ(U) contains the disk . Here is the best possible constant and is referred to as the Koebe constant for S. On the other extreme, ƒ(U) cannot contain the disk {|w| < 1}; unless ƒ is the identity mapping.In order to interpolate between the class S and the identity mapping, one may introduce the families , of functions ƒ ∈ S such that ƒ(U) contains the disk {|w| < d};. Then S(d1) ⊃ S(d2) for d1 < d2, and S(1) contains only the identity mapping. It is obvious that d is the “Koebe constant” for S(d). The relation between d and the second coefficient a2 has been studied by E. Netanyahu [5, 6].

scholarly journals The order of convexity for an integral operator

2016 ◽  
Vol 32 (1) ◽  
pp. 123-129
Author(s):  
VIRGIL PESCAR ◽  
◽  
CONSTANTIN LUCIAN ALDEA ◽  
◽  
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Order Of Convexity

In this paper we consider an integral operator for analytic functions in the open unit disk and we derive the order of convexity for this integral operator, on certain classes of univalent functions.

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 On Sandwich Theorems Results for Certain Univalent Functions Defined by Generalized Operators

2021 ◽  
pp. 2376-2383
Author(s):  
Waggas Galib Atshan ◽  
Aqeel Ahmed Redha Ali
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
Differential Subordination And Superordination

In this present paper, we obtain some differential subordination and superordination results, by using generalized operators for certain subclass of analytic functions in the open unit disk. Also, we derive some sandwich results.

scholarly journals On Differential Sandwich Theorems of Meromorphic Univalent Functions

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Waggas Galib Atshan ◽  
Rajaa Ali Hiress
Keyword(s):  
Linear Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Meromorphic Univalent Functions

        By using of linear  operator, we obtain some Subordinations  and superordinations results for certain normalized meromorphic univalent analytic functions in the in the punctured open unit disk   Also we derive some sandwich theorems .

scholarly journals Symmetric Conformable Fractional Derivative of Complex Variables

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 363 ◽  
Author(s):  
Rabha W. Ibrahim ◽  
Rafida M. Elobaid ◽  
Suzan J. Obaiys
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Unit Disk ◽  
Analytic Functions ◽  
Differential Operators ◽  
Function Theory ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Symmetric Differential Operators

It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to impose two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SCDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk.

scholarly journals On certain analytic univalent functions

2001 ◽  
Vol 25 (5) ◽  
pp. 305-310 ◽  
Author(s):  
B. A. Frasin ◽  
M. Darus
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Strongly Convex ◽  
Strongly Starlike

We consider the class of analytic functionsB(α)to investigate some properties for this class. The angular estimates of functions in the classB(α)are obtained. Finally, we derive some interesting conditions for the class of strongly starlike and strongly convex of orderαin the open unit disk.

scholarly journals Coefficient estimates for certain subclasses of analytic and bi-univalent functions

Filomat ◽  
2015 ◽  
Vol 29 (2) ◽  
pp. 351-360 ◽  
Author(s):  
Yong Sun ◽  
Yue-Ping Jiang ◽  
Antti Rasila
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates

For ? ? 0 and 0 ? ? < 1 < ?, we denote by K(?,?,?) the class of normalized analytic functions satisfying the two sided-inequality ? < K (Zf'(z)/f(z) + z2f''(z)/f(z))<? (z ? U), where U is the open unit disk. Let K? (?, ?, ?) be the class of bi-univalent functions such that f and its inverse f-1 both belong to the class K(?, ?, ?). In this paper, we establish bounds for the coefficients, and solve the Fekete-Szeg? problem, for the class K((?,?,?). Furthermore, we obtain upper bounds for the first two Taylor-Maclaurin coefficients of the functions in the class K? (?,?,?)

scholarly journals Certain subclasses of Spiral-like univalent functions related with Pascal distribution series

2021 ◽  
Vol 7 (2) ◽  
pp. 312-323
Author(s):  
Gangadharan Murugusundaramoorthy
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Inclusion Relations ◽  
Pascal Distribution

Abstract The purpose of the present paper is to find the sufficient conditions for the subclasses of analytic functions associated with Pascal distribution to be in subclasses of spiral-like univalent functions and inclusion relations for such subclasses in the open unit disk 𝔻. Further, we consider the properties of integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

scholarly journals Univalence Conditions of Two New Integral Operators on P-Valent Functions

2019 ◽  
Vol 11 (2) ◽  
pp. 63
Author(s):  
Nguyen Van Tuan ◽  
Daniel Breaz
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
General Integral

For analytic functions in the open unit disk U, we define two new general integral operators. The main object of the this paper is to study these two new integral operators and to determine some sufficient conditions for general p-valent integral operator to be p-th power of a univalent functions.

Coefficient Estimates for a Subclass of Bi-Univalent Functions Defined by q-Derivative Operator

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 306 ◽  
Author(s):  
Suhila Elhaddad ◽  
Maslina Darus
Keyword(s):  
Unit Disk ◽  
Systematic Study ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates ◽  
Derivative Operator ◽  
Coefficient Bounds ◽  
Coefficient Inequalities

Recently, a number of features and properties of interest for a range of bi-univalent and univalent analytic functions have been explored through systematic study, e.g., coefficient inequalities and coefficient bounds. This study examines S q δ ( ϑ , η , ρ , ν ; ψ ) as a novel general subclass of Σ which comprises normalized analytic functions, as well as bi-univalent functions within Δ as an open unit disk. The study locates estimates for the | a 2 | and | a 3 | Taylor–Maclaurin coefficients in functions of the class which is considered. Additionally, links with a number of previously established findings are presented.

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