A geometric criterion for decomposition and multivalence

1988 ◽  
Vol 103 (3) ◽  
pp. 487-495 ◽  
Author(s):  
Y. Abu-Muhanna ◽  
A. Lyzzaik

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.

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Saibah Siregar ◽  
Maslina Darus

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.


Author(s):  
A. Y. Lashin ◽  
F. Z. El-Emam

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.


2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Om P. Ahuja ◽  
Sumit Nagpal ◽  
V. Ravichandran

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.


1998 ◽  
Vol 29 (1) ◽  
pp. 17-28
Author(s):  
KIIALIDA lNAYAT NOOR ◽  
AWATIF A. HENDI

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.


2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Khalida Inayat Noor

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.


2016 ◽  
Vol 95 (2) ◽  
pp. 228-237 ◽  
Author(s):  
MD FIROZ ALI ◽  
A. VASUDEVARAO

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 $f$.


Author(s):  
A. Y. Lashin

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.


1992 ◽  
Vol 23 (4) ◽  
pp. 355-362
Author(s):  
SUBHAS S. TIHOOSNURMATH ◽  
S. R. SWAMY

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.


Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2295-2305
Author(s):  
Ben Wongsaijai ◽  
Nattakorn Sukantamala

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.


1985 ◽  
Vol 32 (3) ◽  
pp. 321-330 ◽  
Author(s):  
K. S. Padmanabhan ◽  
R. Parvatham

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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