- Open Access
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.
- analytic functions
- hypergeometric function
- differential operator
- integral operator
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.
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.
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.
- Alexander JW: Functions which map the interior of the unit circle upon simple regions. Ann. Math. 1915, 17: 12–22. 10.2307/2007212View ArticleGoogle Scholar
- Biernacki M: Sur l’integral des fonctions univalentes. Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 1960, 8: 29–34.MathSciNetGoogle Scholar
- Libera RJ: Some classes of regular univalent functions. Proc. Am. Math. Soc. 1965, 16: 755–758. 10.1090/S0002-9939-1965-0178131-2MathSciNetView ArticleGoogle Scholar
- Bernardi SD: Convex and starlike univalent functions. Trans. Am. Math. Soc. 1969, 135: 429–446.MathSciNetView ArticleGoogle Scholar
- Ruscheweyh S: New criteria for univalent functions. Proc. Am. Math. Soc. 1975, 49: 109–115. 10.1090/S0002-9939-1975-0367176-1MathSciNetView ArticleGoogle Scholar
- Noor KI, Noor MA: On integral operators. J. Math. Anal. Appl. 1999, 238: 341–352. 10.1006/jmaa.1999.6501MathSciNetView ArticleGoogle Scholar
- Noor KI, Noor MA: On certain classes of analytic functions defined by Noor integral operator. J. Math. Anal. Appl. 2003, 281: 244–252. 10.1016/S0022-247X(03)00094-5MathSciNetView ArticleGoogle Scholar
- Noor KI: On new classes of integral operators. J. Nat. Geom. 1999, 16: 71–80.MathSciNetGoogle Scholar
- Kim YC, Srivastava HM: Fractional integral and other linear operators associated with the Gaussian hypergeometric function. Complex Var. Theory Appl. 1997, 34: 293–312. 10.1080/17476939708815054MathSciNetView ArticleGoogle Scholar
- Cho NE: The Noor integral operator and strongly close-to-convex functions. J. Math. Anal. Appl. 2003, 238: 202–212.View ArticleGoogle Scholar
- Cho NE, Kown OS, Srivastava HM: Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators. J. Math. Anal. Appl. 2004, 292: 470–483. 10.1016/j.jmaa.2003.12.026MathSciNetView ArticleGoogle Scholar
- Barnard RW, Kellogg C: Applications of convolution operators to problems in univalent function theory. Mich. Math. J. 1980, 27: 81–94.MathSciNetView ArticleGoogle Scholar
- Livingston A: On the radius of univalence of certain analytic functions. Proc. Am. Math. Soc. 1966, 17: 352–357. 10.1090/S0002-9939-1966-0188423-XMathSciNetView ArticleGoogle Scholar
- Ruscheweyh S: Convolutions in Geometric Function Theory. Les Presses De L’Universite De Montreal, Montreal; 1982.Google Scholar
- Ali RM, Thomas DK: On the starlikeness of the Bernardi integral operator. Proc. Jpn. Acad. 1991, 67: 319–321. 10.3792/pjaa.67.319MathSciNetView ArticleGoogle Scholar
- Oros GI: New results related to the convexity and starlikeness of the Bernardi integral operator. Hacet. J. Math. Stat. 2009, 38: 137–143.MathSciNetGoogle Scholar
- Mogra ML: Applications of Ruscheweyh derivatives and Hadamard product of analytic functions. Int. J. Math. Math. Sci. 1999, 22: 795–805. 10.1155/S0161171299227950MathSciNetView ArticleGoogle Scholar
- Noor KI, Hussain S: On certain analytic functions associated with Ruscheweyh derivatives and bounded Mocanu variation. J. Math. Anal. Appl. 2008, 340: 1145–1152. 10.1016/j.jmaa.2007.09.038MathSciNetView ArticleGoogle Scholar
- Shukla SL, Kumar V: Univalent functions defined by Ruscheweyh derivatives. Int. J. Math. Math. Sci. 1983, 6: 483–486. 10.1155/S0161171283000435MathSciNetView ArticleGoogle Scholar
- Carlson BC, Shaffer DB: Starlike and prestarlike hypergeometric functions. SIAM J. Math. Anal. 1984, 15: 737–745. 10.1137/0515057MathSciNetView ArticleGoogle Scholar
- Shanmugaim TN, Srikandan S, Frasin BA, Kavitha S: On sandwich theorems for certain subclasses of analytic functions involving Carlson-Shaffer operator. J. Korean Math. Soc. 2008, 45: 611–620. 10.4134/JKMS.2008.45.3.611MathSciNetView ArticleGoogle Scholar
- Hohlov YE: Convolution operators preserving univalent functions. Ukr. Math. J. 1985, 37: 220–226.Google Scholar
- Hohlov YE: Hadamard convolution, hypergeometric functions and linear operators in the class of univalent functions. Dokl. Akad. Nauk Ukr. SSR, Ser. A 1984, 7: 25–27.Google Scholar
- Mishra AK, Panigrahi T: Class-mapping properties of the Hohlov operator. Bull. Korean Math. Soc. 2011, 48: 51–65.MathSciNetView ArticleGoogle Scholar
- Owa S, Srivastava HM: Univalent and starlike generalized hypergeometric functions. Can. J. Math. 1987, 39: 1057–1077. 10.4153/CJM-1987-054-3MathSciNetView ArticleGoogle Scholar
- Srivastava HM, Owa S: Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators and certain subclasses of analytic functions. Nagoya Math. J. 1987, 106: 1–28.MathSciNetGoogle Scholar
- Dziok J, Srivastava HM: Classes of analytic functions associated with the generalized hypergeometric function. Appl. Math. Comput. 1999, 103: 1–13.MathSciNetView ArticleGoogle Scholar
- Dziok J, Srivastava HM: Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transforms Spec. Funct. 2003, 14: 7–18. 10.1080/10652460304543MathSciNetView ArticleGoogle Scholar
- Dziok J, Srivastava HM: Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric functions. Adv. Stud. Contemp. Math. 2002, 5: 115–125.MathSciNetGoogle Scholar
- Ali RM, Ravichandran V, Seenivasagan N: Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava linear operator. J. Franklin Inst. 2010, 347: 1762–1781. 10.1016/j.jfranklin.2010.08.009MathSciNetView ArticleGoogle Scholar
- Cho NE, Kwon OS, Ali RM, Ravichandran V: Subordination and superordination for multivalent functions associated with the Dziok-Srivastava operator. J. Inequal. Appl. 2011., 2011: Article ID 486595Google Scholar
- Choi JH, Siago M, Srivastava HM: Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 2002, 276: 432–445. 10.1016/S0022-247X(02)00500-0MathSciNetView ArticleGoogle Scholar
- Ling Y, Liu F: The Choi-Saigo-Srivastava integral operator and a class of analytic functions. Appl. Math. Comput. 2005, 165: 613–621. 10.1016/j.amc.2004.04.031MathSciNetView ArticleGoogle Scholar
- Sokól J: Classes of analytic functions associated with the Choi-Saigo-Srivastava operator. J. Math. Anal. Appl. 2006, 318: 517–525. 10.1016/j.jmaa.2005.06.017MathSciNetView ArticleGoogle Scholar
- Srivastava HM, Attiya AA: An integral operator associated with the Hurwitz-Lerch zeta function and differential subordination. Integral Transforms Spec. Funct. 2007, 18: 207–216. 10.1080/10652460701208577MathSciNetView ArticleGoogle Scholar
- Prajapat JK, Goyal SP: Applications of Srivastava-Attiya operator to the classes of strongly starlike and strongly convex functions. J. Math. Inequal. 2009, 3: 129–137.MathSciNetView ArticleGoogle Scholar
- Al-Oboudi FM, Al-Amoudi KA: On classes of analytic functions related to conic domains. J. Math. Anal. Appl. 2008, 339: 655–667. 10.1016/j.jmaa.2007.05.087MathSciNetView ArticleGoogle Scholar
- Al-Oboudi FM, Al-Amoudi KA: Subordination results for classes of analytic functions related to conic domains defined by a fractional operator. J. Math. Anal. Appl. 2009, 354: 412–420. 10.1016/j.jmaa.2008.10.025MathSciNetView ArticleGoogle Scholar
- Noor KI: On certain classes of p -valent functions. Int. J. Pure Appl. Math. 2005, 18: 345–350.MathSciNetGoogle Scholar
- Aouf MK, El-Ashwah RM: Properties of certain subclasses of meromorphic functions with positive coefficients. Math. Comput. Model. 2009, 49: 868–879. 10.1016/j.mcm.2008.04.013MathSciNetView ArticleGoogle Scholar
- Cho NE, Kim IH: Inclusion properties of certain classes of meromorphic functions associated with the generalized hypergeometric functions. Appl. Math. Comput. 2007, 187: 115–121. 10.1016/j.amc.2006.08.109MathSciNetView ArticleGoogle Scholar
- Raina RK, Sharma P: Harmonic univalent functions associated with Wright’s generalized hypergeometric functions. Integral Transforms Spec. Funct. 2011, 22: 561–572. 10.1080/10652469.2010.535797MathSciNetView ArticleGoogle Scholar
- Ahuja OP: Planar harmonic convolution operators generated by hypergeometric functions. Integral Transforms Spec. Funct. 2007, 18: 165–177. 10.1080/10652460701210227MathSciNetView ArticleGoogle Scholar