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.

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 $\overline{HS}(k,\lambda,b,\alpha)$.

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.

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.

ANALYSIS OF THE TAKENS–BOGDANOV BIFURCATION ON m-PARAMETERIZED VECTOR FIELDS

2010 ◽  
Vol 20 (04) ◽  
pp. 995-1005 ◽  
Author(s):  
FRANCISCO A. CARRILLO ◽  
FERNANDO VERDUZCO ◽  
JOAQUÍN DELGADO
Keyword(s):  
Imaginary Axis ◽  
Vector Fields ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Versal Deformation ◽  
Two Dimensional ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
The Family ◽  
Parameterized Family

Given an m-parameterized family of n-dimensional vector fields, such that: (i) for some value of the parameters, the family has an equilibrium point, (ii) its linearization has a double zero eigenvalue and no other eigenvalue on the imaginary axis, sufficient conditions on the vector field are given such that the dynamics on the two-dimensional center manifold is locally topologically equivalent to the versal deformation of the planar Takens–Bogdanov bifurcation.

scholarly journals Quasiconformal extensions for some geometric subclasses of univalent functions

1984 ◽  
Vol 7 (1) ◽  
pp. 187-195 ◽  
Author(s):  
Johnny E. Brown
Keyword(s):  
Recent Result ◽  
Unit Disk ◽  
Half Plane ◽  
Complex Plane ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Sufficiency Condition ◽  
Extended Complex ◽  
Extended Complex Plane

LetSdenote the set of all functionsfwhich are analytic and univalent in the unit diskDnormalized so thatf(z)=z+a2z2+…. LetS∗andCbe those functionsfinSfor whichf(D)is starlike and convex, respectively. For0≤k<1, letSkdenote the subclass of functions inSwhich admit(1+k)/(1−k)-quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a functionfbelongs toSk⋂S∗orSk⋂C. Functions whose derivatives lie in a half-plane are also considered and a Noshiro-Warschawski-Wolff type sufficiency condition is given to determine which of these functions belong toSk. From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.

scholarly journals On certain classes of close-to-convex functions

1993 ◽  
Vol 16 (2) ◽  
pp. 329-336 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Hadamard Product ◽  
Univalent Functions ◽  
Integral Operators ◽  
Closure Properties ◽  
The Family

A functionf, analytic in the unit diskEand given by ,f(z)=z+∑k=2∞anzkis said to be in the familyKnif and only ifDnfis close-to-convex, whereDnf=z(1−z)n+1∗f,n∈N0={0,1,2,…}and∗denotes the Hadamard product or convolution. The classesKnare investigated and some properties are given. It is shown thatKn+1⫅KnandKnconsists entirely of univalent functions. Some closure properties of integral operators defined onKnare given.

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 The Relationships Between p−valent Functions and Univalent Functions

2015 ◽  
Vol 23 (1) ◽  
pp. 65-72
Author(s):  
Murat Çağlar
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Open Unit ◽  
Open Unit Disk

AbstractIn this paper, we obtain some sufficient conditions for general p−valent integral operators to be the p−th power of a univalent functions in the open unit disk.

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.

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