scholarly journals Generalized Close-to-Convexity Related with Bounded Boundary Rotation

2021 ◽  
Vol 19 (6) ◽  
pp. 890-903
Author(s):  
Khalida Inayat Noor ◽  
Muhammad Aslam Noor ◽  
Muhammad Uzair Awan
Keyword(s):  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Coefficient Bounds ◽  
Special Cases ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Radius Problem

The class Pα,m[A, B] consists of functions p, analytic in the open unit disc E with p(0) = 1 and satisfy p(z) = (m/4 + ½) p1(z) – (m/4 – 1/2) p2(z), m ≥ 2, and p1, p2 are subordinate to strongly Janowski function (1+Az/1+Bz)α, α ∈ (0, 1] and −1 ≤ B < A ≤ 1. The class Pα,m[A, B] is used to define Vα,m[A, B] and Tα,m[A, B; 0; B1], B1 ∈ [−1, 0). These classes generalize the concept of bounded boundary rotation and strongly close-to-convexity, respectively. In this paper, we study coefficient bounds, radius problem and several other interesting properties of these functions. Special cases and consequences of main results are also deduced.

scholarly journals Harmonic univalent functions defined by post quantum calculus operators

2019 ◽  
Vol 11 (1) ◽  
pp. 5-17 ◽  
Author(s):  
Om P. Ahuja ◽  
Asena Çetinkaya ◽  
V. Ravichandran
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Quantum Calculus ◽  
Special Cases ◽  
The Family ◽  
Covering Theorems

Abstract We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

scholarly journals Certain Subclasses of β-Uniformly q-Starlike and β-Uniformly q-Convex Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Eman S. A. AbuJarad ◽  
Mohammed H. A. AbuJarad ◽  
Thabet Abdeljawad ◽  
Fahd Jarad
Keyword(s):  
Convex Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Derivative Operator ◽  
Coefficient Bounds

In this paper, the authors introduced certain subclasses β-uniformly q-starlike and β-uniformly q-convex functions of order α involving the q-derivative operator defined in the open unit disc. Coefficient bounds were also investigated.

scholarly journals Some classes of p-valent analytic functions associated with hypergeometric functions

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1031-1038 ◽  
Author(s):  
Khalida Noor ◽  
Nasir Khan
Keyword(s):  
Linear Operator ◽  
Analytic Functions ◽  
Unit Disc ◽  
Hypergeometric Functions ◽  
Open Unit ◽  
Open Unit Disc ◽  
Class A ◽  
Radius Problem ◽  
Inclusion Results

We define a linear operator on the class A(p) of p-valent analytic functions in the open unit disc involving Gauss hypergeometric functions and introduce certain new subclasses of A(p) using this operator. Some inclusion results, a radius problem and several other interesting properties of these classes are studied.

scholarly journals Coefficient Bounds for Certain Subclasses of Bi-Univalent Function

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
G. Murugusundaramoorthy ◽  
N. Magesh ◽  
V. Prameela
Keyword(s):  
Univalent Function ◽  
Unit Disc ◽  
Univalent Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disc ◽  
Coefficient Bounds

We introduce two new subclasses of the function class Σ of bi-univalent functions defined in the open unit disc. Furthermore, we find estimates on the coefficients|a2|and|a3|for functions in these new subclasses. Also consequences of the results are pointed out.

scholarly journals Coefficient Bounds for Al-Oboudi Type Bi-univalent Functions based on a Modified Sigmoid Activation Function and Horadam Polynomials

2021 ◽  
pp. 251-270
Author(s):  
S. R. Swamy
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Activation Function ◽  
Type Operator ◽  
Open Unit ◽  
Open Unit Disc ◽  
Coefficient Bounds ◽  
Sigmoid Activation Function

Using the Al-Oboudi type operator, we present and investigate two special families of bi-univalent functions in $\mathfrak{D}$, an open unit disc, based on $\phi(s)=\frac{2}{1+e^{-s} },\,s\geq0$, a modified sigmoid activation function (MSAF) and Horadam polynomials. We estimate the initial coefficients bounds for functions of the type $g_{\phi}(z)=z+\sum\limits_{j=2}^{\infty}\phi(s)d_jz^j$ in these families. Continuing the study on the initial cosfficients of these families, we obtain the functional of Fekete-Szeg\"o for each of the two families. Furthermore, we present few interesting observations of the results investigated.

On a certain class of analytic functions associated with a convolution structure

2008 ◽  
Vol 41 (3) ◽  
Author(s):  
G. Murugusundaramoorthy ◽  
R. K. Raina
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Coefficient Bounds ◽  
New Class ◽  
Convolution Structure

AbstractMaking use of a convolution structure, we introduce a new class of analytic functions defined in the open unit disc and investigate its various characteristics. Apart from deriving a set of coefficient bounds, we establish several inclusion relationships involving the (

scholarly journals On q-Uniformly Mocanu Functions

2019 ◽  
Vol 3 (1) ◽  
pp. 5
Author(s):  
Rizwan Badar ◽  
Khalida Noor
Keyword(s):  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Special Cases

Let f be analytic in open unit disc E = { z : | z | < 1 } with f ( 0 ) = 0 and f ′ ( 0 ) = 1 . The q-derivative of f is defined by: D q f ( z ) = f ( z ) - f ( q z ) ( 1 - q ) z , q ∈ ( 0 , 1 ) , z ∈ B - { 0 } , where B is a q-geometric subset of C . Using operator D q , q-analogue class k - U M q ( α , β ) , k-uniformly Mocanu functions are defined as: For k = 0 and q → 1 - , k - reduces to M ( α ) of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases.

Generalized q-starlike functions

2017 ◽  
Vol 54 (4) ◽  
pp. 509-522 ◽  
Author(s):  
Khalida Inayat Noor ◽  
Sadia Riaz
Keyword(s):  
Unit Disc ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disc ◽  
Distortion Theorems ◽  
Bounded Radius Rotation ◽  
Radius Problem

In this paper, we introduce a new concept of q-bounded radius rotation and define the class R*m(q), m ≥ 2, q ∈ (0, 1). The class R*2(q) coincides with S*q which consists of q-starlike functions defined in the open unit disc. Distortion theorems, coefficient result and radius problem are studied. Relevant connections to various known results are pointed out.

Coefficient bounds for M-fold symmetric analytic bi-Bazilevič functions using by Faber polynomial expansion

2020 ◽  
Vol 29 (1) ◽  
pp. 81-89
Author(s):  
F. MUGE SAKAR ◽  
H. OZLEM GUNEY
Keyword(s):  
Unit Disc ◽  
Upper Bounds ◽  
Polynomial Expansion ◽  
Open Unit ◽  
Coefficient Estimates ◽  
Open Unit Disc ◽  
Faber Polynomial ◽  
Coefficient Bounds ◽  
Polynomial Expansions ◽  
Bazilevic Functions

A function is said to be bi-univalent in the open unit disc D, if both the function f and its inverse are univalent in the unit disc. Besides, a function is said to be bi-Bazilevic in ˘ D, if both the function f and its inverse are Bazilevic there. The behaviour of these types of functions are unpredictable ˘ and not much is known about their coefficients. In this study, we determined coefficient estimates for the Taylor Maclaurin coefficients of the class on m-fold symmetric bi-Bazilevic functions. We also, use the Faber Polynomial expansions to obtain these coefficient estimates associated with ˘ upper bounds.

COEFFICIENT BOUNDS FOR A CERTAIN SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS ASSOCIATED WITH THE KOEBE FUNCTION

2018 ◽  
Vol 3 (2) ◽  
pp. 172
Author(s):  
Sidik Bin Rathi ◽  
Shaharuddin Cik Soh ◽  
Ajab Akbarally
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Representation Theorem ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Koebe Function ◽  
Coefficient Bounds

We consider here the functions  which are analytic and univalent in the open unit disc  normalized by   and . By , we denote a new subclass of close-to-convex function such that  for which  and . In this paper, we give the representation theorem and obtain the coefficient bounds for functions in 

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