A Class of Harmonic Starlike Functions defined by Multiplier Transformation

2020 ◽  
pp. 455-469
Author(s):  
Deepali Khurana ◽  
Raj Kumar ◽  
Sibel Yalcin
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Harmonic Mappings ◽  
Distortion Bounds ◽  
Radius Of Convexity ◽  
Univalent Harmonic Mappings ◽  
Harmonic Starlike ◽  
Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

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 Subclasses of Harmonic Mappings Defined by Convolution

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Santosh B. Joshi ◽  
Girish D. Shelake
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Harmonic Mappings ◽  
Coefficient Bounds ◽  
Integral Convolution ◽  
Distortion Bounds ◽  
Special Cases

Two new subclasses of harmonic univalent functions defined by using convolution and integral convolution are introduced. These subclasses generate several known and new subclasses of harmonic univalent functions as special cases and provide a unified treatment in the study of these classes. Coefficient bounds, extreme points, distortion bounds, convolution conditions, and convex combination are also determined.

scholarly journals A Brief Study of Certain Class of Harmonic Functions of Bazilevič Type

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
A. T. Oladipo ◽  
D. Breaz
Keyword(s):  
Linear Operator ◽  
Harmonic Functions ◽  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Distortion Bounds

We define and investigate a new subclass of Bazilevič type harmonic univalent functions using a linear operator. We investigated the harmonic structures in terms of its coefficient conditions, extreme points, distortion bounds, convolution, and convex combination. So, also, we discussed the subordination properties for the functions in this class.

scholarly journals Subclass of Harmonic Univalent Functions Associated with the Generalized Mittag-Leffler Type Functions

2020 ◽  
pp. 139-153
Author(s):  
Adnan Ghazy Alamoush
Keyword(s):  
Unit Disc ◽  
Harmonic Functions ◽  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Distortion Bounds

In the present paper, we introduce a new subclass of harmonic functions in the unit disc U defined by using the generalized Mittag-Leffler type functions. Coefficient conditions, extreme points, distortion bounds, convex combination are studied.

scholarly journals A New Subclass of Harmonic Univalent Functions Associated with Dziok-Srivastava Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
A. L. Pathak ◽  
K. K. Dixit ◽  
R. Agarwal
Keyword(s):  
Integral Operator ◽  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Distortion Bounds

The purpose of the present paper is to study a certain subclass of harmonic univalent functions associated with Dziok-Srivastava operator. We obtain coefficient conditions, distortion bounds, and extreme points for the above class of harmonic univalent 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 combination. The results obtained for the class reduced to the corresponding results for several known classes in the literature are briefly indicated.

scholarly journals A New Subclass of Salagean-Type Harmonic Univalent Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Khalifa Al-Shaqsi ◽  
Maslina Darus ◽  
Olubunmi Abidemi Fadipe-Joseph
Keyword(s):  
Harmonic Functions ◽  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Distortion Bounds

We define and investigate a new subclass of Salagean-type harmonic univalent functions. We obtain coefficient conditions, extreme points, distortion bounds, convolution, and convex combination for the above subclass of harmonic functions.

scholarly journals On Convolution and Convex Combination of Harmonic Mappings

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Ahmad Sulaiman Ahmad El-Faqeer ◽  
Zhen Chuan Ng ◽  
Shamani Supramaniam
Keyword(s):  
Unit Disk ◽  
Imaginary Axis ◽  
Convex Combination ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Horizontal Direction ◽  
Harmonic Mappings ◽  
The Family ◽  
Univalent Harmonic Mappings ◽  
Adequate Condition

In this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings are shown to be convex in the horizontal direction. Also, the family of univalent harmonic mappings of the unit disk onto a region convex in the direction of the imaginary axis is introduced. Sufficient conditions for convex combinations of harmonic mappings of this family to be univalently convex in the direction of the imaginary axis are obtained.

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

scholarly journals Subclasses of Starlike Functions Associated with Fractional q-Calculus Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
G. Murugusundaramoorthy ◽  
C. Selvaraj ◽  
O. S. Babu
Keyword(s):  
Extreme Points ◽  
Starlike Functions ◽  
Partial Sums ◽  
Coefficient Estimate ◽  
Integral Means ◽  
Closure Theorem ◽  
Distortion Bounds

Making use of fractional q-calculus operators, we introduce a new subclass ℳq(λ,γ,k) of starlike functions and determine the coefficient estimate, extreme points, closure theorem, and distortion bounds for functions in ℳq(λ,γ,k). Furthermore we discuss neighborhood results, subordination theorem, partial sums, and integral means inequalities for functions in ℳq(λ,γ,k).

CLASSIFICATION OF UNIVALENT HARMONIC MAPPINGS ON THE UNIT DISK WITH HALF-INTEGER COEFFICIENTS

2014 ◽  
Vol 98 (2) ◽  
pp. 257-280 ◽  
Author(s):  
SAMINATHAN PONNUSAMY ◽  
JINJING QIAO
Keyword(s):  
Unit Disk ◽  
General Result ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Duke Math ◽  
Harmonic Mappings ◽  
Integer Coefficients ◽  
Half Integer ◽  
Univalent Harmonic Mappings

AbstractLet ${\mathcal{S}}$ denote the set of all univalent analytic functions $f$ of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ on the unit disk $|z|<1$. In 1946, Friedman [‘Two theorems on Schlicht functions’, Duke Math. J.13 (1946), 171–177] found that the set ${\mathcal{S}}_{\mathbb{Z}}$ of those functions in ${\mathcal{S}}$ which have integer coefficients consists of only nine functions. In a recent paper, Hiranuma and Sugawa [‘Univalent functions with half-integer coefficients’, Comput. Methods Funct. Theory13(1) (2013), 133–151] proved that the similar set obtained for functions with half-integer coefficients consists of only 21 functions; that is, 12 more functions in addition to these nine functions of Friedman from the set ${\mathcal{S}}_{\mathbb{Z}}$. In this paper, we determine the class of all normalized sense-preserving univalent harmonic mappings $f$ on the unit disk with half-integer coefficients for the analytic and co-analytic parts of $f$. It is surprising to see that there are only 27 functions out of which only six functions in this class are not conformal. This settles the recent conjecture of the authors. We also prove a general result, which leads to a new conjecture.

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