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# Convolution operators in the geometric function theory

- Zahid Shareef
^{1}, - Saqib Hussain
^{2}and - Maslina Darus
^{1}Email author

**2012**:213

https://doi.org/10.1186/1029-242X-2012-213

© Shareef et al.; licensee Springer 2012

**Received:**15 June 2012**Accepted:**13 September 2012**Published:**2 October 2012

## Abstract

The study of operators plays a vital role in mathematics. To define an operator using the convolution theory, and then study its properties, is one of the hot areas of current ongoing research in the geometric function theory and its related fields. In this survey-type article, we discuss historic development and exploit the strengths and properties of some differential and integral convolution operators introduced and studied in the geometric function theory. It is hoped that this article will be beneficial for the graduate students and researchers who intend to start work in this field.

**MSC:**30C45, 30C50.

## Keywords

- analytic functions
- convolution
- hypergeometric function
- differential operator
- integral operator

## 1 Introduction

*A*denote the class of functions of the form

*E*. The convolution or Hadamard product of two functions $f,g\in A$ is denoted by $f\ast g$ and is defined as

where $f(z)$ is given by (1.1), and $g(z)=z+{\sum}_{n=2}^{\mathrm{\infty}}{b}_{n}{z}^{n}$. Note that the convolution of two functions is again analytic in *E*, *i.e.*, $(f\ast g)(z)\in A$.

*a*,

*b*and

*c*with $c\ne 0,-1,-2,-3,\dots $ , the Gauss hypergeometric function ${}_{2}F_{1}(a,b,c;z)$ is defined as

*F*merely signify the number of parameters in the numerator and the denominator of the coefficient of ${z}^{n}$ respectively. Also, ${(\alpha )}_{n}$ is the Pochhammer symbol (or the shifted factorial) defined as

By ${S}^{\ast}$, *C* and *K*, we mean the subclasses of *S* consisting of starlike, with respect to the origin, convex and close-to-convex univalent functions in *E* respectively.

These classes of ${S}^{\ast}$ and *C* are related to each other by the Alexander relation [1, 2]. Later Libera [3] introduced an integral operator and showed that these two classes are closed under this operator. Bernardi [4] gave a generalized operator and studied its properties. Ruscheweyh [5], Noor and Noor [6, 7], Noor [8] and many others, for example, [9–11], defined new operators and studied various classes of analytic and univalent functions generalizing a number of previously known classes and at times discovering new classes of analytic functions.

## 2 Convolution operators

The study of operators plays an important role in the geometric function theory. Many differential and integral operators can be written in terms of convolution of certain analytic functions. It is observed that this formalism brings an ease in further mathematical exploration and also helps to understand the geometric properties of such operators better.

The importance of convolution in the theory of operators may be understood by the following set of examples given by Barnard and Kellogg [12].

This shows how differential and integral operators may be written in terms of convolution of functions. Also, note that once we put these operators into convolution formalism, it becomes easy to conclude that the Alexander differential operator is the inverse of the Alexander integral operator, whereas the Livingston operator is the inverse of the Libera operator. For further discussion on the importance of the convolution operation, we recommend the reader to go through the classical work of Ruscheweyh [14].

Now, we give a brief survey of some convolution operators studied in the geometric function theory in chronological order and mention some related works.

### 2.1 The generalized Bernardi operator (1969)

where ${}_{2}F_{1}(1,1+\eta ,2+\eta ;z)$ is the Gaussian hypergeometric function given by (1.3). The operator ${J}_{\eta}$ was introduced by Bernardi [4]. In [4], it was also shown that the classes ${S}^{\ast}$ and *C* are closed under this operator, *i.e.*, the generalized Bernardi operator maps the classes of ${S}^{\ast}$ and *C* onto the classes of ${S}^{\ast}$ and *C* respectively. Some of other works on the Bernardi operator include [15] and [16] and references therein.

### 2.2 Ruscheweyh derivative operator (1975)

*A*as

*m*th-order Ruscheweyh derivative of $f(z)$. Note that ${D}^{0}f(z)=f(z)$ which is identity operator, and ${D}^{1}f(z)=z{f}^{\mathrm{\prime}}(z)={\mathrm{\Gamma}}_{0}$, the Alexander differential operator. It can also be shown that this operator is hypergeometric in nature as

Many authors, see, for example, [17–19], have used the Ruscheweyh operator to define and investigate the properties of certain known and new classes of analytic functions.

### 2.3 Carlson-Shaffer operator (1984)

*A*onto itself with $L(a,a)$ as the identity if $a\ne 0,-1,-2,\dots $ and $L(c,a)$ for $a\ne 0,-1,-2,\dots $ as the continuous inverse of $L(a,c)$, provided $c\ne 0,-1,-2,\dots $ . Moreover, it is known that

If we take $a=\lambda +1$, $c=1$, then $L(\lambda +1,1)f(z)={D}^{\lambda}f(z)$ which is the Ruscheweyh operator. Therefore, the Carlson-Shaffer operator generalizes the Ruscheweyh derivative operator defined in (2.3). Similarly, we observe that $L(2,1)f(z)={\mathrm{\Gamma}}_{0}$ and $L(3,2)f(z)={\mathrm{\Gamma}}_{1}$.

Recently, Shanmugam *et al.* [21] derived some sandwich theorems of certain subclasses of analytic functions associated with the Carlson-Shaffer operator.

### 2.4 Hohlov linear operator (1984)

and discussed some interesting geometrical properties exhibited by this operator. The three-parameter family of operators ${H}_{a,b,c}$ contains as special cases most of the known linear integral or differential operators. In particular, if $b=1$ in (2.10), then ${H}_{a,1,c}$ reduces to the operator defined in (2.6), implying the Carlson-Shaffer operator a special case of the Hohlov operator. Similarly, it is straightforward to show that the Hohlov operator is also a generalization of Ruscheweyh and Bernardi operators.

*S*to itself for any positive

*ϵ*if $b>0$, $c>{c}_{0}(b)+b+2$, where ${c}_{0}(b)$ is the greatest positive root of the equation

Similarly, an interesting result describing the conditions when the Hohlov linear operator ${H}_{a,b,c}$ maps the class of convex functions *C* to that of univalent functions *S* was also reported in [23]. Recent findings of Mishra *et al.* [24], where they study class-mapping properties of the Hohlov operator, are worth mentioning.

### 2.5 Owa-Srivastava fractional differential operator (1987)

*α*in the sense of Riemann-Liouville is defined as

*i.e.*, $z-\xi >0$, to remove multiplicity of ${(z-\xi )}^{-\alpha}$ in the above integral. Fractional derivatives of higher order are defined by

where $\phi (2,2-\alpha ;z)$ is the incomplete beta function given by (2.7), and Γ denotes the gamma function. Note that ${\mathrm{\Omega}}^{0}=f(z)$, the identity operator, and ${\mathrm{\Omega}}^{1}=z{f}^{\mathrm{\prime}}(z)={\mathrm{\Gamma}}_{0}$, the Alexander differential operator. See [9, 26] for a comprehensive discussion on this operator.

### 2.6 Noor integral operator (1999)

Analogous to Ruscheweyh derivative operator, Noor and Noor [6] and Noor [8] defined an operator as follows.

*n*th-order Noor integral operator. The Noor integral operator in terms of convolution of a hypergeometric function may be given as

The study of the Noor integral operator and its related classes of analytic functions is still a topic of interest for many researchers, *cf.* [7, 10].

### 2.7 Dziok-Srivastava operator (1999)

where $q\le s+1$ and $q,s\in {\mathbb{N}}_{0}$.

For special values of parameters *α*’s, *β*’s, *q* and *s*, we obtain the operators of the generalized Bernardi, Ruscheweyh, Carlson-Shaffer, Hohlov, Owa-Srivastava, Noor and, as we shall see, of Choi-Saigo-Srivastava.

In [30] and [31], the authors have used this operator to produce subordination and superordination results.

### 2.8 Choi-Saigo-Srivastava operator (2002)

In [33] and [34], the authors have discussed some interesting properties of analytic functions associated with the Choi-Saigo-Srivastava operator.

### 2.9 Srivastava-Attiya operator (2007)

where $b\in \mathbb{C}$ with $b\ne 0,-1,-2,-3,\dots $ , $\mu \in \mathbb{C}$, $Re(\mu )>1$ and $|z|<1$.

This operator contains many known operators as its special cases for different values of *μ* and *b*; see [35] for a complete list of such operators. This paper by Srivastava and Attiya discusses some interesting subordination results for this operator as well. Whereas [36] discusses its applications on strongly starlike and convex functions.

### 2.10 The multiplier fractional differential operator (2008)

Al-Oboudi and Al-Amoudi [37] extended the Owa-Srivastava fractional differential operator [25] and proposed a multiplier fractional differential operator.

where ${g}_{\lambda}(z)=\frac{z-(1-\lambda ){z}^{2}}{{(1-z)}^{2}}$. Note that for $n=1$ and $\lambda =0$, ${g}_{\lambda}(z)$ reduces to the identity of a convolution operation, and ${D}_{\lambda}^{n,\alpha}$ becomes the Owa-Srivastava operator (2.12).

In [38], some subordination results have been derived for this fractional operator.

## 3 Concluding remarks

The operation of convolution and convolution operators are the topics of great interest for researchers. Studying operators expressed in terms of convolution helps us explore their geometrical properties. We observe that most of the convolution operators discussed in this paper are hypergeometric operators. We also note that most of the operators discussed in this article are merely special cases of the Hohlov operator which is a special case of the Dziok-Srivastava operator.

Many authors have used these operators on previously known classes of analytic and univalent functions to produce new classes and to investigate several interesting properties of new classes. Also, taking inspiration from these operators, new operators were defined on the classes of multivalent [11, 39], meromorphic [40, 41] and harmonic functions [42, 43].

## Declarations

### Acknowledgements

The work is partially supported by LRGS/TD/2011/UKM/ICT/03/02.

## Authors’ Affiliations

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