scholarly journals Properties of harmonic functions which are convex of order $ \bf \beta $ with respect to symmetric points

2009 ◽  
Vol 40 (1) ◽  
pp. 31-39 ◽  
Author(s):  
Aini Janteng ◽  
Suzeini Abdul Halim
Keyword(s):  
Unit Disc ◽  
Harmonic Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Symmetric Points ◽  
Complex Valued ◽  
Class Of Functions

Let $ \mathcal{H} $ denote the class of functions $ f $ which are harmonic and univalent in the open unit disc $ {D=\{z:|z|<1\}} $. This paper defines and investigates a family of complex-valued harmonic functions that are orientation preserving and univalent in $ \mathcal{D} $ and are related to the functions convex of order $ \beta(0\leq \beta <1) $, with respect to symmetric points. We obtain coefficient conditions, growth result, extreme points, convolution and convex combinations for the above harmonic functions.

scholarly journals A subclass of harmonic univalent functions with positive coefficients

2010 ◽  
Vol 41 (3) ◽  
pp. 261-269 ◽  
Author(s):  
K. K. Dixit ◽  
Saurabh Porwal
Keyword(s):  
Integral Operator ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Distortion Bounds ◽  
Positive Coefficients ◽  
Complex Valued

Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disc $U$ can be written in the form $f=h+\bar g$, where $h$ and $g$ are analytic in $U$. In this paper authors introduce the class, $R_H(\beta)$, $(1<\beta \le 2)$ consisting of harmonic univalent functions $f=h+\bar g$, where $h$ and $g$ are of the form $ h(z)=z+ \sum_{k=2}^\infty |a_k|z^k $ and $ g(z)= \sum_{k=1}^\infty |b_k| z^k $ for which $\Re\{h'(z)+g'(z)\}<\beta$. We obtain distortion bounds extreme points and radii of convexity for functions belonging to this class and discuss a class  preserving integral operator. We also show that class studied in this paper is closed under convolution and convex combinations.

scholarly journals Functions starlike with respect to other points

1991 ◽  
Vol 14 (3) ◽  
pp. 451-456 ◽  
Author(s):  
S. Abdul Halim
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Integral Operators ◽  
Open Unit ◽  
Open Unit Disc ◽  
The Real ◽  
Symmetric Points ◽  
Class Of Functions

In [7], Sakaguchi introduce the class of functions starlike with respect to symmetric points. We extend this class. Forp≤β<1, letSS*(β)be the class of normalised analytic functionsfdefined in the open unit discDsuch thatRezf′(z)/(f(z)−f(−z))>β, for somez ϵ D. In this paper, we introduce 2 other similar classesSC*(β),SSC*(β)as well as give sharp results for the real part of some function forf ϵ SS*(β),SC*(β)andSSC*(β)The behaviour of certain integral operators are also considered.

scholarly journals A Subclass of Harmonic Univalent Functions with Varying Arguments Defined by Generalized Derivative Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
E. A. Eljamal ◽  
M. Darus
Keyword(s):  
Harmonic Functions ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Derivative Operator ◽  
Generalized Derivative ◽  
New Class ◽  
Complex Valued ◽  
Class Of Functions ◽  
Varying Arguments

Making use of the generalized derivative operator, we introduce a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc and are related to uniformly convex functions. We investigate the coefficient bounds, neighborhood, and extreme points for this generalized class of functions.

scholarly journals Subclass of Multivalent Harmonic Functions with Missing Coefficients

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
R. M. El-Ashwah
Keyword(s):  
Unit Disc ◽  
Harmonic Functions ◽  
Extreme Points ◽  
Distortion Theorem ◽  
Open Unit ◽  
Open Unit Disc ◽  
Basic Properties

We have studied subclass of multivalent harmonic functions with missing coefficients in the open unit disc and obtained the basic properties such as coefficient characterization and distortion theorem, extreme points, and convolution.

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 Pascu-Type Harmonic Functions with Positive Coefficients Involving Salagean Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
K. Vijaya ◽  
G. Murugusundaramoorthy ◽  
M. Kasthuri
Keyword(s):  
Harmonic Functions ◽  
Extreme Points ◽  
Partial Sums ◽  
Open Unit ◽  
New Class ◽  
Sălăgean Operator ◽  
Salagean Operator ◽  
Convolution Properties ◽  
Positive Coefficients ◽  
Complex Valued

Making use of a Salagean operator, we introduce a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc. Among the results presented in this paper including the coeffcient bounds, distortion inequality, and covering property, extreme points, certain inclusion results, convolution properties, and partial sums for this generalized class of functions are discussed.

scholarly journals On Certain Subclass of Harmonic Starlike Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
A. Y. Lashin
Keyword(s):  
Integral Operator ◽  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disc ◽  
New Class ◽  
Distortion Bounds ◽  
Harmonic Starlike

Coefficient conditions, distortion bounds, extreme points, convolution, convex combinations, and neighborhoods for a new class of harmonic univalent functions in the open unit disc are investigated. Further, a class preserving integral operator and connections with various previously known results are briefly discussed.

scholarly journals Class of Analytic Function Related with Uniformly Convex and Janowski’s Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Akhter Rasheed ◽  
Saqib Hussain ◽  
Muhammad Asad Zaighum ◽  
Maslina Darus
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Extreme Points ◽  
Distortion Theorem ◽  
Open Unit ◽  
Uniformly Convex ◽  
Coefficient Estimates ◽  
Open Unit Disc

In this paper, we introduce a new subclass of analytic functions in open unit disc. We obtain coefficient estimates, extreme points, and distortion theorem. We also derived the radii of close-to-convexity and starlikeness for this class.

scholarly journals Mapping properties of Janowski-type harmonic functions involving Mittag-Leffler function

2021 ◽  
Vol 6 (12) ◽  
pp. 13235-13246
Author(s):  
Murugusundaramoorthy Gangadharan ◽  
◽  
Vijaya Kaliyappan ◽  
Hijaz Ahmad ◽  
K. H. Mahmoud ◽  
...  
Keyword(s):  
Convolution Operator ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disc

<abstract><p>In this paper, we examine a connotation between certain subclasses of harmonic univalent functions by applying certain convolution operator regarding Mittag-Leffler function. To be more precise, we confer such influences with Janowski-type harmonic univalent functions in the open unit disc $ \mathbb{D}. $</p></abstract>

SUBORDINATION CONDITIONS FOR A CLASS OF NON-BAZILEVIČ TYPE DEFINED BY USING FRACTIONAL Q-CALCULUS OPERATORS

2017 ◽  
pp. 255 ◽  
Author(s):  
S. Abelman ◽  
K. A. Selvakumaran ◽  
M. M. Rashidi ◽  
S. D. Purohit
Keyword(s):  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
New Class ◽  
Symmetric Points ◽  
Bazilevic Functions

In this article, we introduce and investigate a new class of non-Bazilevič functions with respect to k-symmetric points defined by using fractional q-calculus operators and q-differentiation. Several interesting subordination results are derived for the functions belonging to this class in the open unit disc. Furthermore, we point out some new and known consequences of our main result.

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