A geometric criterion for decomposition and multivalence

1988 ◽  
Vol 103 (3) ◽  
pp. 487-495 ◽  
Author(s):  
Y. Abu-Muhanna ◽  
A. Lyzzaik
Keyword(s):  
Univalent Function ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Harmonic Mappings ◽  
Decomposition Property ◽  
Geometric Criterion

AbstractWe give a quite general geometric criterion for a function analytic in the unit disc to be a polynomial of a univalent function, and hence a criterion for multivalence. We believe that this is the essence why multivalent close-to-convex functions enjoy the latter decomposition property. As another application, we study, as suggested by T. Sheil-Small ‘9’, the geometry of classes of analytic functions which arise from his recent investigation of multivalent harmonic mappings.

scholarly journals Certain Conditions for Starlikeness of Analytic Functions of Koebe Type

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Saibah Siregar ◽  
Maslina Darus
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disc ◽  
Subordination Result ◽  
Jack’S Lemma ◽  
Jack's Lemma

For , , we consider the of normalized analytic convex functions defined in the open unit disc . In this paper, we investigate the class , that is, , with is Koebe type, that is, . The subordination result for the aforementioned class will be given. Further, by making use of Jack's Lemma as well as several differential and other inequalities, the authors derived sufficient conditions for starlikeness of the class of -fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated.

A generalized class of α-convex functions with respect to n-symmetric points

2021 ◽  
pp. 2250099
Author(s):  
A. Y. Lashin ◽  
F. Z. El-Emam
Keyword(s):  
Integral Representation ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Symmetric Points

In this paper, we investigate certain subclass of analytic functions on the open unit disc. This class generalizes the well-known class of [Formula: see text]-convex functions with respect to n-symmetric points. Some interesting properties such as subordination results, containment relations, integral preserving properties, and the integral representation for functions in this class are obtained.

scholarly journals Radius Constants for Functions with the Prescribed Coefficient Bounds

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Om P. Ahuja ◽  
Sumit Nagpal ◽  
V. Ravichandran
Keyword(s):  
Unit Disk ◽  
Univalent Function ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Harmonic Mappings ◽  
Coefficient Bounds ◽  
Coefficient Inequality ◽  
Coefficient Bound ◽  
Radius Of Univalence

For an analytic univalent functionf(z)=z+∑n=2∞anznin the unit disk, it is well-known thatan≤nforn≥2. But the inequalityan≤ndoes not imply the univalence off. This motivated several authors to determine various radii constants associated with the analytic functions having prescribed coefficient bounds. In this paper, a survey of the related work is presented for analytic and harmonic mappings. In addition, we establish a coefficient inequality for sense-preserving harmonic functions to compute the bounds for the radius of univalence, radius of full starlikeness/convexity of orderα  (0≤α<1) for functions with prescribed coefficient bound on the analytic part.

scholarly journals ON SOME SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS OF ORDER $\alpha$ TYPE $\delta$

1998 ◽  
Vol 29 (1) ◽  
pp. 17-28
Author(s):  
KIIALIDA lNAYAT NOOR ◽  
AWATIF A. HENDI
Keyword(s):  
Integral Representation ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Starlike Function ◽  
Alpha Type

Let $Q_\lambda^*(\alpha, \delta)$ denote the class of analytic functions $f$ in the unit disc $E$, with $f(0)=0$, $f'(0) =1$ and satisfying the condition \[Re \left\{(1-\lambda)\frac{zf'(z)}{g(z)}+\lambda\frac{(zf'(z))'}{g'(z)}\right\}>\alpha,\] for $z\in E$, $g$ starlike function of order $\delta$ ($0 \le \delta \le 1$), $0\le \alpha \le 1$ and $\lambda$ complex with Re$\lambda\ge 0$. It is shown that $Q_\lambda^*(\alpha, \delta)$ with $\lambda\ge 0$ arc close-to-convex and hence univalent in $E$. Coeffiicient results, an integral representation for $Q_\lambda^*(\alpha, \delta)$ and some other propertie of $Q_\lambda^*(\alpha, \delta)$ are discussed. The class $Q_\lambda^*(\alpha, 1)$ is also investigated in some detail.

scholarly journals Generalizedk-Uniformly Close-to-Convex Functions Associated with Conic Regions

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Integral Operators ◽  
Open Unit ◽  
Open Unit Disc ◽  
Coefficient Problems ◽  
Multiplier Transformation ◽  
Inclusion Results

We define and study some subclasses of analytic functions by using a certain multiplier transformation. These functions map the open unit disc onto the domains formed by parabolic and hyperbolic regions and extend the concept of uniformly close-to-convexity. Some interesting properties of these classes, which include inclusion results, coefficient problems, and invariance under certain integral operators, are discussed. The results are shown to be the best possible.

scholarly journals LOGARITHMIC COEFFICIENTS OF SOME CLOSE-TO-CONVEX FUNCTIONS

2016 ◽  
Vol 95 (2) ◽  
pp. 228-237 ◽  
Author(s):  
MD FIROZ ALI ◽  
A. VASUDEVARAO
Keyword(s):  
Univalent Function ◽  
Upper Bound ◽  
Convex Functions ◽  
Unit Disc ◽  
Starlike Functions ◽  
Logarithmic Coefficients

The logarithmic coefficients$\unicode[STIX]{x1D6FE}_{n}$of an analytic and univalent function$f$in the unit disc$\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$with the normalisation$f(0)=0=f^{\prime }(0)-1$are defined by$\log (f(z)/z)=2\sum _{n=1}^{\infty }\unicode[STIX]{x1D6FE}_{n}z^{n}$. In the present paper, we consider close-to-convex functions (with argument 0) with respect to odd starlike functions and determine the sharp upper bound of$|\unicode[STIX]{x1D6FE}_{n}|$,$n=1,2,3$, for such functions $f$.

scholarly journals Convolution conditions for bounded \(\alpha\)-starlike functions of complex order

2017 ◽  
Vol 71 (1) ◽  
pp. 65
Author(s):  
A. Y. Lashin
Keyword(s):  
Complex Plane ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
General Class ◽  
Starlike Functions ◽  
Starlike And Convex Functions ◽  
Complex Order ◽  
Sălăgean Operator ◽  
Salagean Operator

Let \(A\) be the class of analytic functions in the unit disc \(U\) of the complex plane \(\mathbb{C}\) with the normalization \(f(0)=f^{^{\prime }}(0)-1=0\). We introduce a subclass \(S_{M}^{\ast }(\alpha ,b)\) of \(A\), which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class \(S_{M}^{\ast }(n,\alpha ,b)\) (\(n\geq 0\)) related to \(S_{M}^{\ast }(\alpha ,b)\) is also considered under the same conditions. Among other things, we find convolution conditions for a function \(f\in A\) to belong to the class \(S_{M}^{\ast }(\alpha ,b)\). Several properties of the class \(S_{M}^{\ast }(n,\alpha ,b)\) are investigated.

scholarly journals ROTARU ALPHA - CONVEX FUNCTIONS

1992 ◽  
Vol 23 (4) ◽  
pp. 355-362
Author(s):  
SUBHAS S. TIHOOSNURMATH ◽  
S. R. SWAMY
Keyword(s):  
Real Number ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Representation Formula ◽  
Negative Real Number

Let $S^*(a,b)$ denote the class of analytic functions $f$ in the unit disc $E$, with $f(0) =f'(0) - 1 =0$, satisfying the condition $|(zf'(z)/f(z))- a|<b$, $a\in C$, $|a- 1|<b\le Re(a)$, $z\in E$. In this paper the class $S^*(\alpha, a, b)$ of functions $f$ analytic in $E$, with $f(0) = f'(0)- 1 =0$, $f(z)f'(z)/z\neq 0$ for $z$ in $E$ and satisfying in $E$ the condition $|J(\alpha,f)- a|<b$, $a \in C$, $|a-1|<b\le Re(a)$, where $J(\alpha, f) =(1- \alpha)(zf'(z)/f(z)) +\alpha((zf'(z))'/f'(z))$, $\alpha$ a non-negative real number is introduced. It is proved that $S^*(\alpha, a,b)\subset S^*(a,b)$, if $a> (4b/c)|Im(a)|$, $c=(b^2- |a- 1|^2)/b$. Further a representation formula for $f \in S^*(\alpha, a, b)$ and an inequality relating the coefficients of functions in $S^*(\alpha, a, b)$ are obtained.

scholarly journals A certain class of q-close-to-convex functions of order α

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2295-2305
Author(s):  
Ben Wongsaijai ◽  
Nattakorn Sukantamala
Keyword(s):  
Hypergeometric Function ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disc ◽  
Bieberbach Conjecture ◽  
Gaussian Hypergeometric Function ◽  
Geometric Properties

For every 0 < q < 1 and 0 ? ? < 1, we introduce a class of analytic functions f on the open unit disc D with the standard normalization f(0)= 0 = f'(0)-1 and satisfying |1/1-?(z(Dqf)(z)/h(z)-?)- 1/1-q,(z?D), where h?S*q. This class is denoted by Kq(?), so called the class of q-close-to-convex-functions of order ?. In this paper, we study some geometric properties of this class. In addition, we consider the famous Bieberbach conjecture problem on coefficients for the class Kq(?). We also find some sufficient conditions for the function to be in Kq(?) for some particular choices of the functions h. Finally, we provide some applications on q-analogue of Gaussian hypergeometric function.

scholarly journals Some applications of differential subordination

1985 ◽  
Vol 32 (3) ◽  
pp. 321-330 ◽  
Author(s):  
K. S. Padmanabhan ◽  
R. Parvatham
Keyword(s):  
Univalent Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Hadamard Product ◽  
Differential Subordination ◽  
Convex Univalent Function

Let Sa (h) denote the class of analytic functions f on the unit disc E with f (0) =0 = f′ (0) −1 satisfying , where (a real), denotes the Hadamard product of Ka with f, and h is a convex univalent function on E, with Re h > 0. Let . It is proved that F ε Sa (h) whenever f ε Sa (h) and also that for a ≥ 1. Three more such classes are introduced and studied here. The method of differential subordination due to Eenigenburg et al. is used.

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