Abstract and Applied Analysis

Volume 2014, Article ID 640856, 6 pages

http://dx.doi.org/10.1155/2014/640856

## Initial Coefficients of Biunivalent Functions

^{1}School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia^{2}Department of Mathematics, University of Delhi, Delhi 110 007, India

Received 24 January 2014; Accepted 12 March 2014; Published 8 April 2014

Academic Editor: Om P. Ahuja

Copyright © 2014 See Keong Lee 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

An analytic function defined on the open unit disk is biunivalent if the function and its inverse are univalent in . Estimates for the initial coefficients of biunivalent functions are investigated when and , respectively, belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.

#### 1. Introduction

Let be the class of all univalent analytic functions in the open unit disk and normalized by the conditions and . For , it is well known that the th coefficient is bounded by . The bounds for the coefficients give information about the geometric properties of these functions. Indeed, the bound for the second coefficient of functions in the class gives rise to the growth, distortion and covering theorems for univalent functions. In view of the influence of the second coefficient in the geometric properties of univalent functions, it is important to know the bounds for the (initial) coefficients of functions belonging to various subclasses of univalent functions. In this paper, we investigate this coefficient problem for certain subclasses of biunivalent functions.

Recall that the Koebe one-quarter theorem [1] ensures that the image of under every univalent function contains a disk of radius 1/4. Thus, every univalent function has an inverse satisfying , , and
A function is* biunivalent* in if both and are univalent in . Let denote the class of biunivalent functions defined in the unit disk . Lewin [2] investigated this class and obtained the bound for the second coefficient of the biunivalent functions. Several authors subsequently studied similar problems in this direction (see [3, 4]). A function is bistarlike or strongly bistarlike or biconvex of order if and are both starlike, strongly starlike, or convex of order , respectively. Brannan and Taha [5] obtained estimates for the initial coefficients of bistarlike, strongly bistarlike, and biconvex functions. Bounds for the initial coefficients of several classes of functions were also investigated in [6–24].

An analytic function is* subordinate* to an analytic function , written , if there is an analytic function with satisfying . Ma and Minda [25] unified various subclasses of starlike () and convex functions () by requiring that either the quantity or is subordinate to a more general superordinate function with positive real part in the unit disk , , , maps onto a region starlike with respect to and symmetric with respect to the real axis. The class of Ma-Minda starlike functions with respect to consists of functions satisfying the subordination . Similarly, the class of Ma-Minda convex functions consists of functions satisfying the subordination . Ma and Minda investigated growth and distortion properties of functions in and as well as Fekete-Szegö inequalities for and . Their proof of Fekete-Szegö inequalities requires the univalence of . Ali et al. [7] investigated Fekete-Szegö problems for various other classes and their proof does not require the univalence or starlikeness of . In particular, their results are valid even if one just assumes the function to have a series expansion of the form , . So, in this paper, we assume that has series expansion , , are real, and . A function is Ma-Minda bistarlike or Ma-Minda biconvex if both and are, respectively, Ma-Minda starlike or convex. Motivated by the Fekete-Szegö problem for the classes of Ma-Minda starlike and Ma-Minda convex functions [25], Ali et al. [26] recently obtained estimates of the initial coefficients for biunivalent Ma-Minda starlike and Ma-Minda convex functions.

The present work is motivated by the results of Kędzierawski [27] who considered functions belonging to certain subclasses of univalent functions while their inverses belong to some other subclasses of univalent functions. Among other results, he obtained the following coefficient estimates.

Theorem 1 (see [27]). *Let with Taylor series and . Then,
*

*We need the following classes investigated in [6, 7, 26].*

*Definition 2. *Let be analytic and with and . For , let

*In this paper, we obtain the estimates for the second and third coefficients of functions when(i) and , or , or ,(ii) and , or ,(iii) and .*

*2. Coefficient Estimates*

*In the sequel, it is assumed that and are analytic functions of the form
*

*Theorem 3. Let and . If , and is of the form
then
where .*

*Proof. *Since and , , then there exist analytic functions , with , satisfying
Define the functions and by
or, equivalently,
Then, and are analytic in with . Since , the functions and have positive real part in , and and . In view of (8) and (10), it is clear that
Using (10) together with (4), it is evident that
Since has the Maclaurin series given by (5), a computation shows that its inverse has the expansion
Since
it follows from (11) and (12) that
It follows from (15) and (17) that
Equations (15), (16), (18), and (19) lead to
where , which, in view of and , gives us the desired estimate on as asserted in (6).

By using (16), (18), and (19), we get
and this yields the estimate given in (7).

*Remark 4. *When and , , then (6) reduces to Theorem 1. When and , Theorem 3 reduces to [26, Theorem 2.2].

*Theorem 5. Let and . If and , then
where .*

*Proof. *Let and , . Then, there exist analytic functions , with , such that
Since
(12) and (24) yield
It follows from (26) and (28) that
Hence, (26), (27), (29), and (30) lead to
which gives us the desired estimate on as asserted in (22) when and .

Further, (27), (29), and (30) give
and this yields the estimate given in (23).

*Theorem 6. Let and . If and , then
where .*

*Proof. *Let and , . Then, there are analytic functions , with , satisfying
Using
and (12) and (34) will yield
Further implication of (36) and applying the fact that and give the estimates in (33).

*Theorem 7. Let and . If , , then
where .*

*Proof. *For and , , there exist analytic functions , with , satisfying
Since
then (12) and (38) yield
Further implication of (40) and applying the fact that and give the estimates in (37).

*Remark 8. *When and , Theorem 7 reduces to [26, Theorem 2.3].

*The following theorems give the estimates for the second and third coefficients of functions when (i) and and (ii) and . The proofs are similar as for the theorems above; hence, they are omitted here.*

*Theorem 9. Let and . If and , then
where .*

*Theorem 10. Let and . If and , then
where .*

*Remark 11. *When and , Theorem 10 reduces to [26, Theorem 2.4].

*Conflict of Interests*

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

*Acknowledgments*

*The research of the first and last authors is supported, respectively, by FRGS Grant and MyBrain MyPhD Programme of the Ministry of Higher Education, Malaysia.*

*References*

- P. L. Duren,
*Univalent Functions*, vol. 259, Springer, New York, NY, USA, 1983. 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 - 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 - E. Netanyahu, “The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $\left|z\right|<1$,”
*Archive for Rational Mechanics and Analysis*, vol. 32, pp. 100–112, 1969. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. A. Brannan and T. S. Taha, “On some classes of bi-univalent functions,”
*Universitatis Babeş-Bolyai. Studia. Mathematica*, vol. 31, no. 2, pp. 70–77, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. M. Ali, S. K. Lee, V. Ravichandran, and S. Supramaniam, “The Fekete-Szegő coefficient functional for transforms of analytic functions,”
*Iranian Mathematical Society. Bulletin*, vol. 35, no. 2, article 276, pp. 119–142, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. M. Ali, V. Ravichandran, and N. Seenivasagan, “Coefficient bounds for $p$-valent functions,”
*Applied Mathematics and Computation*, vol. 187, no. 1, pp. 35–46, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - 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 - A. K. Mishra and P. Gochhayat, “Fekete-Szegö problem for a class defined by an integral operator,”
*Kodai Mathematical Journal*, vol. 33, no. 2, pp. 310–328, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. N. Shanmugam, C. Ramachandran, and V. Ravichandran, “Fekete-Szegő problem for subclasses of starlike functions with respect to symmetric points,”
*Bulletin of the Korean Mathematical Society*, vol. 43, no. 3, pp. 589–598, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. Srivastava, “Some inequalities and other results associated with certain subclasses of univalent and bi-univalent analytic functions,” in
*Nonlinear Analysis*, vol. 68 of*Springer Series on Optimization and Its Applications*, pp. 607–630, Springer, Berlin, Germany, 2012. View at Publisher · 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, 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 - 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 - G. Murugusundaramoorthy, N. Magesh, and V. Prameela, “Coefficient bounds for certain subclasses of bi-univalent function,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 573017, 3 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - H. Tang, G.-T. Deng, and S.-H. Li, “Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions,”
*Journal of Inequalities and Applications*, vol. 2013, article 317, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. G. Hamidi, S. A. Halim, and J. M. Jahangiri, “Coefficent estimates for bi-univalent strongly starlike and Bazilevic functions,”
*International Journal of Mathematics Research*, vol. 5, no. 1, pp. 87–96, 2013. View at Google Scholar - S. Bulut, “Coefficient estimates for initial Taylor-Maclaurin coefficients for a subclass of analytic and bi-univalent functions defined by Al-Oboudi differential operator,”
*The Scientific World Journal*, vol. 2013, Article ID 171039, 6 pages, 2013. View at Publisher · View at Google Scholar - S. Bulut, “Coefficient estimates for a class of analytic and bi-univalent functions,”
*Novi Sad Journal of Mathematics*, vol. 43, no. 2, pp. 59–65, 2013. View at Google Scholar - N. Magesh, T. Rosy, and S. Varma, “Coefficient estimate problem for a new subclass of biunivalent functions,”
*Journal of Complex Analysis*, vol. 2013, Article ID 474231, 3 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - H. M. Srivastava, G. Murugusundaramoorthy, and N. Magesh, “On certain subclasses of bi-univalent functions associated with Hohlov operator,”
*Global Journal of Mathematical Analysis*, vol. 1, no. 2, pp. 67–73, 2013. View at Google Scholar - M. Çağlar, H. Orhan, and N. Yağmur, “Coefficient bounds for new subclasses of bi-univalent functions,”
*Filomat*, vol. 27, no. 7, pp. 1165–1171, 2013. View at Publisher · View at Google Scholar - H. M. Srivastava, S. Bulut, M. C. Çağlar, and N. Yağmur, “Coefficient estimates for a general subclass of analytic and bi-univalent functions,”
*Filomat*, vol. 27, no. 5, pp. 831–842, 2013. View at Publisher · View at Google Scholar - S. S. Kumar, V. Kumar, and V. Ravichandran, “Estimates for the initial coefficients of bi-univalent functions,”
*Tamsui Oxford Journal of Information and Mathematical Science*. In press. - W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” in
*Proceedings of the Conference on Complex Analysis (Tianjin, 1992)*, Conference Proceedings and Lecture Notes in Analysis, pp. 157–169, International Press, Cambridge, Mass, USA. - 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 - A. W. Kędzierawski, “Some remarks on bi-univalent functions,”
*Annales Universitatis Mariae Curie-Skłodowska. Section A. Mathematica*, vol. 39, no. 1985, pp. 77–81, 1988. View at Google Scholar · View at MathSciNet

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