scholarly journals Planar harmonic mappings in a family of functions convex in one direction

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 431-445
Author(s):  
Sudhananda Maharan ◽  
Swadesh Sahoo
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Harmonic Mapping ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Optimal Estimates ◽  
Convex In One Direction ◽  
Planar Harmonic Mappings ◽  
Class Of Functions

Let D := {z ? C : |z| < 1} be the open unit disk, and h and 1 be two analytic functions in D. Suppose that f = h + ?g is a harmonic mapping in D with the usual normalization h(0) = 0 = g(0) and h'(0) = 1. In this paper, we consider harmonic mappings f by restricting its analytic part to a family of functions convex in one direction and, in particular, starlike. Some sharp and optimal estimates for coefficient bounds, growth, covering and area bounds are investigated for the class of functions under consideration. Also, we obtain optimal radii of fully convexity, fully starlikeness, uniformly convexity, and uniformly starlikeness of functions belonging to those family.

scholarly journals Composition Operators on Some Banach Spaces of Harmonic Mappings

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Munirah Aljuaid ◽  
Flavia Colonna
Keyword(s):  
Banach Spaces ◽  
Unit Disk ◽  
Besov Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Zygmund Space ◽  
Spaces Of Analytic Functions

We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of analytic functions on the open unit disk in the complex plane, including the α-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.

scholarly journals Some properties of multivalent analytic functions associated with an integral operator

2013 ◽  
Vol 21 (1) ◽  
pp. 277-284
Author(s):  
Yi-Hui Xu ◽  
Cai-Mei Yan
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Class Of Functions

Abstract Let A(p) denote the class of functions of the form f(z) = zp Σ∞k=1+p akzk (p ∈ N = {1, 2, 3,...}) which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1} By making use of the Noor integral operator, we obtain some interesting properties of multivalent analytic functions.

scholarly journals A Generalized Class of Functions Defined by the q-Difference Operator

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2361
Author(s):  
Loriana Andrei ◽  
Vasile-Aurel Caus
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
New Class ◽  
The Relationship ◽  
Class Of Functions

The goal of the present investigation is to introduce a new class of analytic functions (Kt,q), defined in the open unit disk, by means of the q-difference operator, which may have symmetric or assymetric properties, and to establish the relationship between the new defined class and appropriate subordination. We derived relationships of this class and obtained sufficient conditions for an analytic function to be Kt,q. Finally, in the concluding section, we have taken the decision to restate the clearly-proved fact that any attempt to create the rather simple (p,q)-variations of the results, which we have provided in this paper, will be a rather inconsequential and trivial work, simply because the added parameter p is obviously redundant.

scholarly journals Fractional Calculus of Analytic Functions Concerned with Möbius Transformations

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Nicoleta Breaz ◽  
Daniel Breaz ◽  
Shigeyoshi Owa
Keyword(s):  
Fractional Calculus ◽  
Unit Disk ◽  
Analytic Functions ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Open Unit ◽  
Open Unit Disk ◽  
Möbius Transformations ◽  
Class Of Functions ◽  
Mobius Transformations

LetAbe the class of functionsf(z)in the open unit diskUwithf(0)=0andf′(0)=1. Also, letw(ζ)be a Möbius transformation inUfor somez∈U. Applying the Möbius transformations, we consider some properties of fractional calculus (fractional derivatives and fractional integrals) off(z)∈A. Also, some interesting examples for fractional calculus are given.

scholarly journals On a class of analytic functions governed by subordination

2019 ◽  
Vol 11 (1) ◽  
pp. 144-155 ◽  
Author(s):  
Ravinder Krishna Raina ◽  
Janusz Sokół
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Basic Properties ◽  
Radius Of Convexity ◽  
Class Of Functions

Abstract The purpose of this paper is to introduce a class of functions ℱλ, λ ∈ [0, 1], consisting of analytic functions f normalized by f(0) = f´(0) − 1 = 0 in the open unit disk U which satisfies the subordination condition that $${\rm{z}}f'\left( {\rm{z}} \right)/\left\{ {\left( {1 - \lambda } \right){\rm{f}}\left( {\rm{z}} \right) + \lambda {\rm{z}}} \right\} \prec {\rm{q}}\left( {\rm{z}} \right),\,\,\,\,\,{\rm{z}} \in {\rm{\mathbb{U},}}$$ where ${\rm{q}}\left( {\rm{z}} \right) = \sqrt {1 + {{\rm{z}}^{\rm{2}}}} + {\rm{z}}$ . Some basic properties (including the radius of convexity) are obtained for this class of functions.

scholarly journals Some properties of a class of analytic functions

2018 ◽  
Vol 37 (4) ◽  
pp. 173-186
Author(s):  
Neng Xu ◽  
R. K. Raina
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Special Cases ◽  
Class Of Functions

Making use of convolution, we introduce and investigate a certain class of functions which is analytic in the open unit disk. We obtain interesting properties of starlikeness and convexity for this function class. Special cases and some useful consequences of our main results are also mentioned.

scholarly journals A Class of Analytic Functions with Missing Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Ding-Gong Yang ◽  
Jin-Lin Liu
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates ◽  
Sharp Bounds ◽  
Radius Of Convexity ◽  
Convolution Properties ◽  
Class Of Functions

Let and denote the class of functions of the form which are analytic in the open unit disk and satisfy the following subordination condition , for, for. We obtain sharp bounds on , and coefficient estimates for functions belonging to the class . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.

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 Partial sums of certain analytic functions

2001 ◽  
Vol 25 (12) ◽  
pp. 771-775 ◽  
Author(s):  
Shigeyoshi Owa
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Partial Sums ◽  
Open Unit ◽  
Open Unit Disk

The object of the present paper is to consider the starlikeness and convexity of partial sums of certain analytic functions in the open unit disk.

scholarly journals The order of convexity for an integral operator

2016 ◽  
Vol 32 (1) ◽  
pp. 123-129
Author(s):  
VIRGIL PESCAR ◽  
◽  
CONSTANTIN LUCIAN ALDEA ◽  
◽  
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Order Of Convexity

In this paper we consider an integral operator for analytic functions in the open unit disk and we derive the order of convexity for this integral operator, on certain classes of univalent functions.

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