Further results for two certain subclasses of close-to-convex functions

2020 ◽  
pp. 2150045
Author(s):  
H. Mahzoon ◽  
R. Kargar
Keyword(s):  
Convex Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
The Family ◽  
Radius Of Univalence

Let [Formula: see text] be the family of analytic and normalized functions [Formula: see text] in the open unit disc [Formula: see text]. In this paper, we consider the following classes: [Formula: see text] and [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. We show that if [Formula: see text], then [Formula: see text] and [Formula: see text] are greater than [Formula: see text], and if [Formula: see text], then [Formula: see text]. Also, some another interesting properties of the class [Formula: see text] are investigated. Finally, the radius of univalence of 2nd section sum of [Formula: see text] is obtained.

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.

Majorization-Subordination Theorems for Locally Univalent Functions, II

1973 ◽  
Vol 25 (2) ◽  
pp. 420-425 ◽  
Author(s):  
Douglas Michael Campbell
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disc ◽  
The Family ◽  
Linear Invariant ◽  
Linear Invariant Family

Let denote the set of all normalized analytic univalent functions in the open unit disc D. Let f(z), F(z) and φ(z) be analytic in |z| < r. We say that f(z) is majorized by F(z) in we say that f(z) is subordinate to F(z) in where .Let be the set of all locally univalent (f’(z) ≠ 0) analytic functions in D with order ≦α which are of the form f(z) = z +… . The family is known as the universal linear invariant family of order α [6]. A concise summary of and introduction to properties of linear invariant families which relate to the following material is contained in [1]. The present paper contains the proofs of some of the results announced in [1]

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 Certain Conditions for Starlikeness of Analytic Functions of Koebe Type

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Saibah Siregar ◽  
Maslina Darus
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disc ◽  
Subordination Result ◽  
Jack’S Lemma ◽  
Jack's Lemma

For , , we consider the of normalized analytic convex functions defined in the open unit disc . In this paper, we investigate the class , that is, , with is Koebe type, that is, . The subordination result for the aforementioned class will be given. Further, by making use of Jack's Lemma as well as several differential and other inequalities, the authors derived sufficient conditions for starlikeness of the class of -fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated.

A generalized class of α-convex functions with respect to n-symmetric points

2021 ◽  
pp. 2250099
Author(s):  
A. Y. Lashin ◽  
F. Z. El-Emam
Keyword(s):  
Integral Representation ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Symmetric Points

In this paper, we investigate certain subclass of analytic functions on the open unit disc. This class generalizes the well-known class of [Formula: see text]-convex functions with respect to n-symmetric points. Some interesting properties such as subordination results, containment relations, integral preserving properties, and the integral representation for functions in this class are obtained.

scholarly journals Radius Problems for Starlike Functions Associated with the Tan Hyperbolic Function

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Khalil Ullah ◽  
Saira Zainab ◽  
Muhammad Arif ◽  
Maslina Darus ◽  
Meshal Shutaywi
Keyword(s):  
Holomorphic Function ◽  
Convex Functions ◽  
Unit Disc ◽  
Starlike Functions ◽  
Open Unit ◽  
Hyperbolic Function ◽  
Open Unit Disc ◽  
Starlike And Convex Functions ◽  
Radius Problems

The aim of this particular article is at studying a holomorphic function f defined on the open-unit disc D = z ∈ ℂ : z < 1 for which the below subordination relation holds z f ′ z / f z ≺ q 0 z = 1 + tan h z . The class of such functions is denoted by S tan h ∗ . The radius constants of such functions are estimated to conform to the classes of starlike and convex functions of order β and Janowski starlike functions, as well as the classes of starlike functions associated with some familiar functions.

scholarly journals Coefficient inequality for transforms of starlike and convex functions

2015 ◽  
Vol 24 (1) ◽  
pp. 69-75
Author(s):  
D. VAMSHEE KRISHNA ◽  
◽  
B. VENKATESWARLU ◽  
T. RAMREDDY ◽  
◽  
...  
Keyword(s):  
Analytic Function ◽  
Complex Plane ◽  
Upper Bound ◽  
Convex Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Toeplitz Determinants ◽  
Starlike And Convex Functions ◽  
Coefficient Inequality

The objective of this paper is to obtain an upper bound for the second Hankel functional associated with the k th root transform ... normalized analytic function f(z) belonging to starlike and convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.

scholarly journals Generalizedk-Uniformly Close-to-Convex Functions Associated with Conic Regions

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Integral Operators ◽  
Open Unit ◽  
Open Unit Disc ◽  
Coefficient Problems ◽  
Multiplier Transformation ◽  
Inclusion Results

We define and study some subclasses of analytic functions by using a certain multiplier transformation. These functions map the open unit disc onto the domains formed by parabolic and hyperbolic regions and extend the concept of uniformly close-to-convexity. Some interesting properties of these classes, which include inclusion results, coefficient problems, and invariance under certain integral operators, are discussed. The results are shown to be the best possible.

scholarly journals Certain Subclasses of Bi-Close-to-Convex Functions Associated with Quasi-Subordination

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Gurmeet Singh ◽  
Gurcharanjit Singh ◽  
Gagandeep Singh
Keyword(s):  
Convex Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc

In the present investigation, we introduce certain new subclasses of the class of biunivalent functions in the open unit disc U=z:z<1 defined by quasi-subordination. We obtained estimates on the initial coefficients a2 and a3 for the functions in these subclasses. The results present in this paper would generalize and improve those in related works of several earlier authors.

scholarly journals A certain class of q-close-to-convex functions of order α

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2295-2305
Author(s):  
Ben Wongsaijai ◽  
Nattakorn Sukantamala
Keyword(s):  
Hypergeometric Function ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disc ◽  
Bieberbach Conjecture ◽  
Gaussian Hypergeometric Function ◽  
Geometric Properties

For every 0 < q < 1 and 0 ? ? < 1, we introduce a class of analytic functions f on the open unit disc D with the standard normalization f(0)= 0 = f'(0)-1 and satisfying |1/1-?(z(Dqf)(z)/h(z)-?)- 1/1-q,(z?D), where h?S*q. This class is denoted by Kq(?), so called the class of q-close-to-convex-functions of order ?. In this paper, we study some geometric properties of this class. In addition, we consider the famous Bieberbach conjecture problem on coefficients for the class Kq(?). We also find some sufficient conditions for the function to be in Kq(?) for some particular choices of the functions h. Finally, we provide some applications on q-analogue of Gaussian hypergeometric function.

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