Journal of Complex Analysis

Volume 2014, Article ID 693908, 6 pages

http://dx.doi.org/10.1155/2014/693908

## Coefficient Estimate of Biunivalent Functions of Complex Order Associated with the Hohlov Operator

^{1}Faculty of Mathematics and Computer Science, Hubei University, Wuhan 430062, China^{2}School of Advanced Sciences, VIT University, Vellore, Tamilnadu 632014, India

Received 2 January 2014; Accepted 2 March 2014; Published 10 April 2014

Academic Editor: J. Dziok

Copyright © 2014 Z. Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce and investigate a new subclass of the function class of biunivalent functions of complex order defined in the open unit disk, which are associated with the Hohlov operator, satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients and for functions in this new subclass. Several, known or new, consequences of the results are also pointed out.

#### 1. Introduction, Definitions, and Preliminaries

Let denote the class of functions of the following form: which are analytic in the open unit disk By we denote the class of all functions in which are univalent in . Some of the important and well-investigated subclasses of the class include, for example, the class of starlike functions of order in and the class of convex functions of order in . It is well known that every function has an inverse , defined by where

A function is said to be biunivalent in , if and are univalent in . Let denote the class of biunivalent functions in given by (1).

An analytic function is subordinate to an analytic function , written , provided that there is an analytic function defined on with and satisfying . Ma and Minda [1] unified various subclasses of starlike and convex functions for which either of the quantity or is subordinate to a more general superordinate function. For this purpose, they considered an analytic function with positive real part in the unit disk , , , and maps onto a region starlike with respect to 1 and symmetric with respect to the real axis. The class of Ma-Minda starlike functions consists of functions satisfying the subordination . Similarly, the class of Ma-Minda convex functions consists of functions satisfying the subordination .

A function is bi-starlike of Ma-Minda type or biconvex of Ma-Minda type, if both and are, respectively, Ma-Minda starlike or convex. These classes are denoted, respectively, by and . In the sequel, it is assumed that is an analytic function with positive real part in the unit disk , satisfying and , and is symmetric with respect to the real axis. Such a function has a series expansion of the form

The convolution or Hadamard product of two functions and is denoted by and is defined as where is given by (1) and . Here, in our present investigation, we recall a convolution operator due to Hohlov [2, 3], which indeed is a special case of the Dziok-Srivastava operator [4, 5].

For the complex parameters , , and , the Gaussian hypergeometric function is defined as where is the Pochhammer symbol (or the shifted factorial) defined as follows: For the positive real values , , and , by using the Gaussian hypergeometric function given by (7), Hohlov [2, 3] introduced the familiar convolution operator as follows: where Hohlov [2, 3] discussed some interesting geometrical properties exhibited by the operator . The three-parameter family of operators contains, as its special cases, most of the known linear integral or differential operators. In particular, if in (9), then reduces to the Carlson-Shaffer operator. Similarly, it is easily seen that the Hohlov operator is also a generalization of the Ruscheweyh derivative operator as well as the Bernardi-Libera-Livingston operator.

Recently, there has been triggering interest to study biunivalent function class and obtained nonsharp coefficient estimates on the first two coefficients and of (1). But the coefficient problem for each of the Taylor-Maclaurin coefficients, is still an open problem (see [6–11]). Many researchers (see [12–17]) have recently introduced and investigated several interesting subclasses of the biunivalent function class and they have found nonsharp estimates on the first two Taylor-Maclaurin coefficients and .

Motivated by the earlier work of Deniz [18] (see [19–21]) and Peng and Han [22], in the present paper, we introduce new subclasses of the function class of complex order , involving Hohlov operator , and find estimates on the coefficients and for functions in the new subclasses of function class . Several related classes are also considered, and connection to earlier known results are made.

*Definition 1. *A function given by (1) is said to be in the class , if the following conditions are satisfied:
where the function is given by (4).

On specializing the parameters and , , and , one can state the various new subclasses of as illustrated in the following examples.

*Example 2. *For and , a function , given by (1), is said to be in the class , if the following conditions are satisfied:
where , and the function is given by (4).

*Example 3. *For and , a function , given by (1), is said to be in the class , if the following conditions are satisfied:
where , and the function is given by (4).

It is of interest to note that, for and , the class reduces to the following new subclasses.

*Example 4. *For and , a function , given by (1), is said to be in the class , if the following conditions are satisfied:
where , and the function is given by (4).

*Example 5. *For and , a function , given by (1), is said to be in the class , if the following conditions are satisfied:
where and the function is given by (4).

In the following section, we find estimates on the coefficients and for functions in the above-defined subclasses of the function class by employing the technique which is different from that used by earlier authors. Earlier authors investigated the coefficients of biunivalent functions mainly by using the following lemma.

Lemma 6 (see [23]). *If , then for each , where is the family of all functions , analytic in , for which
**
where
*

*2. Coefficient Bounds for the Function Class *

*We begin by finding the estimates on the coefficients and for functions in the class .*

*Suppose that and are analytic in with , , and and suppose that
It is well known that
Thus, from (5), it follows that
*

*Theorem 7. Let a function , given by (1), be in the class . Thenwhere and are given by (10).*

*Proof. *It follows from (12) that
where and are given by (21) and (22), respectively.

Now, by equating the coefficients in (24), we get
From (25) and (27), we find that
which implies
By adding (26) and (28) and by using (29) and (30), we obtain
Now, by using (20) and (31), we get
Hence,
This gives the bound on as asserted in (23).

Next, in order to find the bound on , by subtracting (28) from (26), we get
It follows from (20), (30), and (35) that
By using (34), we obtainThis completes the proof of Theorem 7.

*By putting in Theorem 7, we have the following corollary.*

*Corollary 8. Let the function given by (1) be in the class . Then
*

*By taking and , in Corollary 8, we get the following corollary.*

*Corollary 9. Let the function given by (1) be in the class . Then
*

*By putting in Theorem 7, we have the following corollary.*

*Corollary 10. Let the function given by (1) be in the class . Then*

*By taking and , in Corollary 10, we get the following corollary.*

*Corollary 11. Let the function given by (1) be in the class . Then
*

*3. Concluding Remarks *

*3. Concluding Remarks**For the class of strongly starlike functions, the function is given by
which gives and .*

*Remark 12. *From Theorem 7, when and for the class [8], we get

On the other hand, if we take
then .

*Remark 13. *From Theorem 7, when for the class , we get

*Remark 14. *By putting in Corollary 11 we obtain more accurate results corresponding to the results obtained in [19]. Further, by taking and is given by (43) (or by (45), the results obtained in Theorem 7 and Corollary 11 yield more accurate results than the results obtained in [15, 21].

*Remark 15. *If , , and with , then the operator turns into well-known Bernardi operator:
and are the well-known Alexander and Libera operators, respectively. Further, if in (9), then immediately yields the Carlson-Shaffer operator . So, various other interesting corollaries and consequences of our main results (which are asserted by Theorem 7 above) can be derived similarly. The details involved may be left as an exercise for the interested reader.

*Conflict of Interests*

*Conflict of Interests**The authors declare that there is no conflict of interests regarding the publication of this paper.*

*References*

*References*

- W. C. Ma and D. Minda, “A unified treatment of some special classes of functions,” in
*Proceedings of the Conference on Complex Analysis, Tianjin, 1992, 157–169*, vol. 1 of*Conference Proceedings and Lecture Notes in Analysis*, International Press, Cambridge, Mass, USA, 1994. View at Google Scholar - Y. E. Khokhlov, “Convolution operators that preserve univalent functions,”
*Ukrainskiĭ Matematicheskiĭ Zhurnal*, vol. 37, no. 2, pp. 220–226, 1985. View at Google Scholar · View at MathSciNet - Y. E. Khokhlov, “Hadamard convolutions, hypergeometric functions and linear operators in the class of univalent functions,”
*Doklady Akademiya Nauk Ukrainskoĭ SSR. A. Fiziko-Matematicheskie i Tekhnicheskie Nauki*, no. 7, pp. 25–27, 1984. View at Google Scholar · View at MathSciNet - J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hypergeometric function,”
*Applied Mathematics and Computation*, vol. 103, no. 1, pp. 1–13, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Dziok and H. M. Srivastava, “Certain subclasses of analytic functions associated with the generalized hypergeometric function,”
*Integral Transforms and Special Functions*, vol. 14, no. 1, pp. 7–18, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. A. Brannan, J. Clunie, and W. E. Kirwan, “Coefficient estimates for a class of star-like functions,”
*Canadian Journal of Mathematics*, vol. 22, pp. 476–485, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. A. Brannan and J. G. Clunie,
*Aspects of Contemporary Complex Analysis*, Academic Press, London, UK, 1980. - D. A. Brannan and T. S. Taha, “On some classes of bi-univalent functions,”
*Studia Universitatis Babeş-Bolyai Mathematica*, vol. 31, no. 2, pp. 70–77, 1986. View at Google Scholar · View at MathSciNet - M. Lewin, “On a coefficient problem for bi-univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 18, pp. 63–68, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Netanyahu, “The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $zx3c;\phantom{\rule{0.166667em}{0ex}}1$,”
*Archive for Rational Mechanics and Analysis*, vol. 32, pp. 100–112, 1969. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. S. Taha,
*Topics in univalent function theory [Ph.D. thesis]*, University of London, London, UK, 1981. - B. A. Frasin and M. K. Aouf, “New subclasses of bi-univalent functions,”
*Applied Mathematics Letters*, vol. 24, no. 9, pp. 1569–1573, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Hayami and S. Owa, “Coefficient bounds for bi-univalent functions,”
*Panamerican Mathematical Journal*, vol. 22, no. 4, pp. 15–26, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. F. Li and A. P. Wang, “Two new subclasses of bi-univalent functions,”
*International Mathematical Forum*, vol. 7, no. 29–32, pp. 1495–1504, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. Srivastava, A. K. Mishra, and P. Gochhayat, “Certain subclasses of analytic and bi-univalent functions,”
*Applied Mathematics Letters*, vol. 23, no. 10, pp. 1188–1192, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. H. Xu, Y. C. Gui, and H. M. Srivastava, “Coefficient estimates for a certain subclass of analytic and bi-univalent functions,”
*Applied Mathematics Letters*, vol. 25, no. 6, pp. 990–994, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. H. Xu, H. G. Xiao, and H. M. Srivastava, “A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems,”
*Applied Mathematics and Computation*, vol. 218, no. 23, pp. 11461–11465, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Deniz, “Certain subclasses of bi-univalent functions satisfying subordinate conditions,”
*Journal of Classical Analysis*, vol. 2, no. 1, pp. 49–60, 2013. View at Google Scholar - R. M. Ali, S. K. Lee, V. Ravichandran, and S. Supramaniam, “Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions,”
*Applied Mathematics Letters*, vol. 25, no. 3, pp. 344–351, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Panigrahi and G. Murugusundaramoorthy, “Coefficient bounds for bi-univalent analytic functions associated with Hohlov operator,”
*Proceedings of the Jangjeon Mathematical Society*, vol. 16, no. 1, pp. 91–100, 2013. View at Google Scholar · View at MathSciNet - H. M. Srivastava, G. Murugusundaramoorthy, and N. Magesh, “Certain subclasses of bi-univalent functions associated with the Hohlov operator,”
*Global Journal of Mathematical Analysis*, vol. 1, no. 2, pp. 67–73, 2013. View at Google Scholar - Z. Peng and Q. Han, “On the coefficients of several classes of bi-univalent functions,”
*Acta Mathematica Scientia B*, vol. 34, no. 1, pp. 228–240, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - C. Pommerenke,
*Univalent Functions*, Vandenhoeck & Ruprecht, Göttingen, Germany, 1975. View at MathSciNet

*
*