scholarly journals Convolutions of prestarlike functions

1983 ◽  
Vol 6 (1) ◽  
pp. 59-68 ◽  
Author(s):  
O. P. Ahuja ◽  
H. Silverman
Keyword(s):  
Unit Disk ◽  
Extremal Function ◽  
Extreme Points ◽  
Distortion Theorems ◽  
The Extremal Function ◽  
Class Of Functions

The convolution of two functionsf(z)=∑n=0∞anznandg(z)=∑n=0∞bnzndefined as(f∗g)(z)=∑n=0∞anbnzn. Forf(z)=z−∑n=2∞anznandg(z)=z/(1−z)2(1−γ), the extremal function for the class of functions starlike of orderγ, we investigate functionsh, whereh(z)=(f∗g)(z), which satisfy the inequality|(zh′/h)−1|/|(zh′/h)+(1-2α)|<β,0≤α<1,0<β≤1for allzin the unit disk. Such functionsfare said to beγ-prestarlike of orderαand typeβ. We characterize this family in terms of its coefficients, and then determine extreme points, distortion theorems, and radii of univalence, starlikeness, and convexity. All results are sharp.

SOME NEW RESULTS ASSOCIATED WITH CARATHÉODORY FUNCTIONS OF ORDER α

2016 ◽  
Vol 2 (1) ◽  
pp. 11-13
Author(s):  
H. M. Srivastava ◽  
Shigeyoshi Owa
Keyword(s):  
Unit Disk ◽  
Extremal Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
New Class ◽  
Carathéodory Functions ◽  
Coefficient Inequalities ◽  
The Extremal Function ◽  
Class Of Functions

Let $\mathcal{P}(\alpha)$ be the class of functions $p(z)$ which are Carathéodory functions of order $\alpha$ $(0 \le\alpha<1)$ in the open unit disk $\mathbf{U}$. Considering the extremal function $p(z)$ for the class $\mathcal{P}(\alpha)$, a new class ${\mathcal P}^*(\beta)$ $\beta\in \mathbb{R}$ of functions $q(z)$ in $\mathbf{U}$ is defined. The object of the present paper is to develop several interesting coefficient inequalities for the functions $q(z)$ in the new class $\mathcal{P}^*(\beta)$ introduced here.

scholarly journals On a maximal outer area problem for a class of meromorphic univalent fuctions

1986 ◽  
Vol 34 (3) ◽  
pp. 433-445 ◽  
Author(s):  
Stephen M. Zemyan
Keyword(s):  
Unit Disk ◽  
Extremal Function ◽  
Additional Information ◽  
Area Problem ◽  
Class Of Functions ◽  
Outer Area

For 0 < p < 1, let Sp denote the class of functions f (z) which are meromorphic and univalent in the unit disk U, with the normalisations f (0) = 0, f′(0) = 1 and f (p) = ∞, and let Sp (a) denote subclass of Sp consisting of those functions in Sp whose residue at the pole in equal to a. In this paper, we determine, for values of the residue a in a certain disk Δp, the greatest possible outer area over all functions in the class Sp (a). We also determine additional information concerning extremal function if the reside a dose not lie in Δp.

New classes of certain analytic functions concerned with subordinations

2017 ◽  
Vol 33 (2) ◽  
pp. 153-160
Author(s):  
NICOLETA BREAZ ◽  
◽  
SHIGEYOSHI OWA ◽  
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Extremal Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
The Extremal Function

Let A be the class of analytic functions f(z) in the open unit disk U which satisfy f(0) = 0 and f 0(0) = 1. Applying the extremal function for the subclass S∗(α) of A, new classes P∗(α) and Q∗(α) are considered using certain subordinations. The object of the present paper is to discuss some interesting properties for f(z) belonging to the classes P∗(α) and Q∗(α)

scholarly journals A NEW SUBCLASS OF STARLIKE HARMONIC FUNCTIONS DEFINED BY SUBORDINATION

2020 ◽  
Vol 25 (1) ◽  
Author(s):  
Serkan Çakmak ◽  
Sibel Yalçın ◽  
Şahsene Altınkaya
Keyword(s):  
Harmonic Functions ◽  
Extreme Points ◽  
Current Work ◽  
Coefficient Bounds ◽  
Distortion Theorems ◽  
Class Of Functions

In this current work, by using a relation of subordination, we define a new subclass of starlike harmonic functions. We get coefficient bounds, distortion theorems, extreme points, convolution and convex combinations for this class of functions. Moreover, some relevant connections of the results presented here with diverse known results are briefly denoted.

scholarly journals Inequalities of harmonic univalent functions with connections of hypergeometric functions

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Janusz Sokół ◽  
Rabha W. Ibrahim ◽  
M. Z. Ahmad ◽  
Hiba F. Al-Janaby
Keyword(s):  
Unit Disk ◽  
Hypergeometric Functions ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Class Of Functions

AbstractLet SH be the class of functions f = h+g that are harmonic univalent and sense-preserving in the open unit disk U = { z : |z| < 1} for which f (0) = f'(0)-1=0. In this paper, we introduce and study a subclass H( α, β) of the class SH and the subclass NH( α, β) with negative coefficients. We obtain basic results involving sufficient coefficient conditions for a function in the subclass H( α, β) and we show that these conditions are also necessary for negative coefficients, distortion bounds, extreme points, convolution and convex combinations. In this paper an attempt has also been made to discuss some results that uncover some of the connections of hypergeometric functions with a subclass of harmonic univalent functions.

Coefficient inequalities related with typically real functions

2020 ◽  
Vol 70 (4) ◽  
pp. 829-838
Author(s):  
Saqib Hussain ◽  
Shahid Khan ◽  
Khalida Inayat Noor ◽  
Mohsan Raza
Keyword(s):  
Unit Disk ◽  
Real Functions ◽  
Coefficient Inequalities ◽  
Typically Real ◽  
Class Of Functions ◽  
Typically Real Functions

AbstractIn this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.

scholarly journals On the univalency for certain subclass of analytic functions involving Ruscheweyh derivatives

2002 ◽  
Vol 31 (9) ◽  
pp. 567-575 ◽  
Author(s):  
Liu Mingsheng
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Covering Theorem ◽  
Inclusion Relations ◽  
Special Case ◽  
Class Of Functions

LetHbe the class of functionsf(z)of the formf(z)=z+∑ K=2 + ∞a k z k, which are analytic in the unit diskU={z;|z|<1}. In this paper, we introduce a new subclassBλ(μ,α,ρ)ofHand study its inclusion relations, the condition of univalency, and covering theorem. The results obtained include the related results of some authors as their special case. We also get some new results.

Certain Class of Meromorphically Multivalent Functions Associated with a Linear Operator

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Dr. Jitendra Awasthi
Keyword(s):  
Linear Operator ◽  
Unit Disk ◽  
Primary 30C45 ◽  
Closure Properties ◽  
Distortion Theorems ◽  
Ams Classification

In this paper we have introduced a new subclass ( , , , , , ) ,, T a c A B p m l    of meromorphic p-valent functions in the punctured unit disk U*={z∈C:0 less than |z| less than 1} by sing a linear operator p m l I , , .Coefficients inequalities, Distortion theorems, Closure properties, Radii of starlikeness and convexity are obtained for this class. 2010 AMS CLASSIFICATION: Primary 30C45.

scholarly journals On the asymptotic boundary behavior of functions analytic in the unit disk

1977 ◽  
Vol 67 ◽  
pp. 1-13
Author(s):  
James R. Choike
Keyword(s):  
Riemann Surface ◽  
Analytic Function ◽  
Unit Disk ◽  
Boundary Behavior ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Asymptotic Boundary ◽  
Necessary And Sufficient ◽  
Class Of Functions

In [8] a necessary and sufficient condition was given for determining the equivalence of two asymptotic boundary paths for an analytic function w = f(p) on a Riemann surface F. In this paper we give a necessary and sufficient condition for determining the nonequivalence of two asymptotic boundary paths for f(z) analytic in |z| < R, 0 < R ≤ + ∞. We shall, also, illustrate some applications of the main result and examine a class of functions introduced by Valiron.

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