scholarly journals Extreme Points Method and Univalent Harmonic Mappings

2016 ◽  
pp. 223-237 ◽  
Author(s):  
Yusuf Muhanna ◽  
Saminathan Ponnusamy
Keyword(s):  
Extreme Points ◽  
Harmonic Mappings ◽  
Univalent Harmonic Mappings

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.

scholarly journals Improved Bohr’s inequality for locally univalent harmonic mappings

2019 ◽  
Vol 30 (1) ◽  
pp. 201-213 ◽  
Author(s):  
Stavros Evdoridis ◽  
Saminathan Ponnusamy ◽  
Antti Rasila
Keyword(s):  
Harmonic Mappings ◽  
Bohr’S Inequality ◽  
Univalent Harmonic Mappings

scholarly journals Coefficients of univalent harmonic mappings

2017 ◽  
Vol 186 (3) ◽  
pp. 453-470 ◽  
Author(s):  
Saminathan Ponnusamy ◽  
Anbareeswaran Sairam Kaliraj ◽  
Victor V. Starkov
Keyword(s):  
Harmonic Mappings ◽  
Univalent Harmonic Mappings

On a subclass of p-harmonic mappings

2013 ◽  
Vol 44 (3) ◽  
pp. 313-325 ◽  
Author(s):  
Saurabh Porwal ◽  
Kaushal Kishore Dixit
Keyword(s):  
Extreme Points ◽  
Harmonic Mapping ◽  
Harmonic Mappings

The purpose of the present paper is to introduce two new classes $HS_p(\alpha)$ and $HC_p(\alpha)$ of $p$-harmonic mappings together with their corresponding subclasses $HS^0_p(\alpha)$ and $HC^0_p(\alpha)$. We prove that the mappings in $HS_p(\alpha)$ and $HC_p(\alpha)$ are univalent and sense-preserving in $U$ and obtain extreme points of $HS^0_p(\alpha)$ and $HC^0_p(\alpha)$, $HS_p(\alpha)\cap T_p$ and $HC_p(\alpha)\cap T_p$ are determined, where $T_p$ denotes the set of $p$-harmonic mapping with non negative coefficients. Finally, we establish the existence of the neighborhoods of mappings in $HC_p(\alpha)$. Relevant connections of the results presented here with various known results are briefly indicated.

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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