A SPECIAL CLASS OF UNIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS

2020 ◽  
Vol 27 (1) ◽  
pp. 1-13
Author(s):  
M. K. Aouf
Keyword(s):  
Unit Disc ◽  
Special Class ◽  
Univalent Functions ◽  
Alpha Beta ◽  
Special Classes

There are many special classes of univalent functions in the unit disc U. In this paper, we consider the special class P(A, B, alpha, beta, -1 <= B < A <= 1, - 1 <= B <= 0, 0 < =alpha < 1 and 0 < beta <= 1, of univalent functions in the unit disc U. And it is the purpose of this paper to show some properties of this class.

scholarly journals A SPECIAL CLASS OF UNIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS

1996 ◽  
Vol 27 (1) ◽  
pp. 1-13
Author(s):  
M. K. AOUF
Keyword(s):  
Unit Disc ◽  
Special Class ◽  
Univalent Functions ◽  
Alpha Beta ◽  
Special Classes

There are many special classes of univalent functions in the unit disc $U$. In this paper, we consider the special class $P^*(A,B,\alpha,\beta)$, $-1\le B<A \le 1$, $-1 \le B < 0$, $0 \le\alpha < 1$ and $0 < \beta\le 1$, of univalent functions mthe umt disc $U$. And it is the purpose of this paper to show some properties of this class.

scholarly journals Certain properties of rational functions involving Bessel functions

2012 ◽  
Vol 43 (3) ◽  
pp. 391-398
Author(s):  
Rasoul Aghalary ◽  
Ali Ebadian ◽  
Zahra Oroujy
Keyword(s):  
Analytic Function ◽  
Bessel Function ◽  
Unit Disc ◽  
Bessel Functions ◽  
Univalent Functions ◽  
Rational Functions ◽  
Alpha Beta

Let $g_{\upsilon}(z)$ be the classical Bessel function of the first kind of order $\upsilon$ and $f$ be an analytic function defined on the unit disc $\Delta$. Suppose the operator $H(f)$ be defined by $H(f)(z)=\frac{z}{\frac{z}{f(z)}*\frac{g_{\upsilon}(z)}{z}}$. In this paper we identify subfamily $M_{n}(\alpha,\beta)$ of univalent functions and obtain conditions on the parameter $\upsilon$ such that $f\in M_{n}(\alpha,\beta)$ implies $H(f)\in M_{n}(\alpha,\beta)$.

scholarly journals A GENERALIZATION OF CERTAIN CLASSES OF ANALYTIC FUNCTIONS

1994 ◽  
Vol 25 (1) ◽  
pp. 41-51
Author(s):  
M. K. AOUF
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Coefficient Estimates ◽  
Alpha Beta ◽  
Distortion Theorems ◽  
Special Classes

There are many classes of analytic functions in the unit disc $U$. We consider about the special classes $S^*_\lambda(A,B,\alpha,\beta)$ and $C^*_\lambda(A,B,\alpha,\beta)$( $-1\le A < B \le 1$, $0 < B \le 1$, $0 \le \alpha < 1$ and $0 < \beta \le 1$) of analytic functions in the unit disc $U$. And the purpose of this paper is to study the classes $S^*_\lambda(A,B,\alpha,\beta)$ and $C^*_\lambda(A,B,\alpha,\beta)$. We prove some distortion theorems and some coefficient estimates for these classes $S^*_\lambda(A,B,\alpha,\beta)$ and $C^*_\lambda(A,B,\alpha,\beta)$.

scholarly journals ON CERTAIN SUBCLASS OF UNIVALENT FUNCTIONS IN THE UNIT DISC I

1995 ◽  
Vol 26 (4) ◽  
pp. 299-312
Author(s):  
M. K. AOUF ◽  
A. SHAMANDY ◽  
M. F. YASSEN
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Integral Operators ◽  
Coefficient Estimates ◽  
Closure Theorems ◽  
Alpha Beta ◽  
Distortion Theorems ◽  
Analytic And Univalent Functions

The object of the present paper is to derive several interesting proper- ties of the class $P_n(\alpha, \beta, \gamma)$ consisting of analytic and univalent functions with neg- ative coefficients. Coefficient estimates, distortion theorems and closure theorems of functions in the class $P_n(\alpha, \beta, \gamma)$ are determined. Also radii of close-to-convexity, starlikeness and convexity and integral operators are determined.

Hankel determinants of second and third order for the class 𝓢 of univalent functions

2021 ◽  
Vol 71 (3) ◽  
pp. 649-654
Author(s):  
Milutin Obradović ◽  
Nikola Tuneski
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Third Order ◽  
Hankel Determinants

Abstract In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class 𝓢 of univalent functions in the unit disc.

scholarly journals Harmonic univalent functions defined by post quantum calculus operators

2019 ◽  
Vol 11 (1) ◽  
pp. 5-17 ◽  
Author(s):  
Om P. Ahuja ◽  
Asena Çetinkaya ◽  
V. Ravichandran
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Quantum Calculus ◽  
Special Cases ◽  
The Family ◽  
Covering Theorems

Abstract We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

Agencies Referring Children for Special Class Placement

1981 ◽  
Vol 5 (1) ◽  
pp. 15-16
Author(s):  
Diane L. Bowyer ◽  
E. Constable
Keyword(s):  
Special Education ◽  
Teacher Training ◽  
Young Children ◽  
Special Class ◽  
Classroom Teachers ◽  
Class Placement ◽  
Detailed Assessment ◽  
Training Courses ◽  
Special Classes

AbstractThe present study investigated the sources of referral of young children placed in Junior Special Classes. It was found that more than half of the children were referred by kindergarten or classroom teachers. These results were discussed in the light of (i) overseas findings; (ii) the need for special education content in teacher training courses; and, (iii) providing practising teachers with a checklist for ascertaining which children require detailed assessment.

Formulas for the Nehari Coefficients of Bounded Univalent Functions

1977 ◽  
Vol 29 (3) ◽  
pp. 587-605
Author(s):  
Duane W. De Temple ◽  
David B. Oulton
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Bieberbach Conjecture ◽  
Grunsky Inequalities

The Grunsky inequalities [6] and their generalizations (e.g., [5; 14; 17]) have become an increasingly important tool for the study of the coefficients of normalized univalent functions defined on the unit disc. In particular, proofs based upon the Grunsky inequalities have now settled the Bieberbach conjecture for the fifth [15] and sixth [13] coefficients. For bounded univalent functions the situation is similar, although the Grunsky inequalities go over to those of Nehari [11].

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.

Majorization-Subordination Theorems for Locally Univalent Functions, II

1973 ◽  
Vol 25 (2) ◽  
pp. 420-425 ◽  
Author(s):  
Douglas Michael Campbell
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disc ◽  
The Family ◽  
Linear Invariant ◽  
Linear Invariant Family

Let denote the set of all normalized analytic univalent functions in the open unit disc D. Let f(z), F(z) and φ(z) be analytic in |z| < r. We say that f(z) is majorized by F(z) in we say that f(z) is subordinate to F(z) in where .Let be the set of all locally univalent (f’(z) ≠ 0) analytic functions in D with order ≦α which are of the form f(z) = z +… . The family is known as the universal linear invariant family of order α [6]. A concise summary of and introduction to properties of linear invariant families which relate to the following material is contained in [1]. The present paper contains the proofs of some of the results announced in [1]

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