Convolution operators in the geometric function theory
© Shareef et al.; licensee Springer 2012
Received: 15 June 2012
Accepted: 13 September 2012
Published: 2 October 2012
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.
where is given by (1.1), and . Note that the convolution of two functions is again analytic in E, i.e., .
By , 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 and C are related to each other by the Alexander relation [1, 2]. Later Libera  introduced an integral operator and showed that these two classes are closed under this operator. Bernardi  gave a generalized operator and studied its properties. Ruscheweyh , Noor and Noor [6, 7], Noor  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 .
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 .
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 is the Gaussian hypergeometric function given by (1.3). The operator was introduced by Bernardi . In , it was also shown that the classes and C are closed under this operator, i.e., the generalized Bernardi operator maps the classes of and C onto the classes of and C respectively. Some of other works on the Bernardi operator include  and  and references therein.
2.2 Ruscheweyh derivative operator (1975)
2.3 Carlson-Shaffer operator (1984)
If we take , , then which is the Ruscheweyh operator. Therefore, the Carlson-Shaffer operator generalizes the Ruscheweyh derivative operator defined in (2.3). Similarly, we observe that and .
Recently, Shanmugam et al.  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 contains as special cases most of the known linear integral or differential operators. In particular, if in (2.10), then 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.
Similarly, an interesting result describing the conditions when the Hohlov linear operator maps the class of convex functions C to that of univalent functions S was also reported in . Recent findings of Mishra et al. , where they study class-mapping properties of the Hohlov operator, are worth mentioning.
2.5 Owa-Srivastava fractional differential operator (1987)
where is the incomplete beta function given by (2.7), and Γ denotes the gamma function. Note that , the identity operator, and , the Alexander differential operator. See [9, 26] for a comprehensive discussion on this operator.
2.6 Noor integral operator (1999)
2.7 Dziok-Srivastava operator (1999)
where and .
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.
2.8 Choi-Saigo-Srivastava operator (2002)
2.9 Srivastava-Attiya operator (2007)
where with , , and .
This operator contains many known operators as its special cases for different values of μ and b; see  for a complete list of such operators. This paper by Srivastava and Attiya discusses some interesting subordination results for this operator as well. Whereas  discusses its applications on strongly starlike and convex functions.
2.10 The multiplier fractional differential operator (2008)
where . Note that for and , reduces to the identity of a convolution operation, and becomes the Owa-Srivastava operator (2.12).
In , 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].
The work is partially supported by LRGS/TD/2011/UKM/ICT/03/02.
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