Journal of Complex Analysis
Volume 2017 (2017), Article ID 2826514, 9 pages
https://doi.org/10.1155/2017/2826514

Research Article

Coefficients Bounds for Certain Subclass of Biunivalent Functions Associated with Ruscheweyh -Differential Operator

Saqib Hussain,¹ Shahid Khan,² Muhammad Asad Zaighum,² Maslina Darus,³ and Zahid Shareef4,5

¹Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan
²Department of Mathematics, Riphah International University, Islamabad, Pakistan
³School of Mathematical Sciences, Faculty of Sciences and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
⁴Foundation Program, Dhofar University, Salalah, Oman
⁵Division of Engineering, Higher Colleges of Technology, P.O. Box 4114, Fujairah, UAE

Correspondence should be addressed to Saqib Hussain; saqib_math@yahoo.com

Received 16 May 2017; Accepted 27 June 2017; Published 5 September 2017

Academic Editor: Yan Xu

Copyright © 2017 Saqib Hussain 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 in our present investigation a new subclass of analytic and biunivalent functions associated with Ruscheweyh -differential operator in open unit disk . We use the Faber polynomial expansions to find th coefficients bounds of class of bisubordinate functions and also find initial coefficient estimates.

1. Introduction

Let denote the class of all function which is analytic in the open unit disk and has the Taylor series expansion of the following form:By we mean the class of all functions in which are univalent in The Koebe one-quarter theorem [1] states that the image of under every function from contains a disk of radius It is well known that every univalent function has an inverse which is defined as whereA function is said to be biunivalent in if both and are univalent in .

If the functions and are analytic in , then is said to be subordinate to , written as if there exists a Schwarz function , analytic in , with and , such that Let denote the class of analytic and biunivalent functions in given by the Taylor-Maclaurin series expansion (1). Here we give few examples of functions in the class such that However, the famous Koebe function is not in ; for more details we refer to [2]. For , the class of biunivalent analytic functions was first introduced and studied by Lewin where it was proved to show that . Brannan and Clunie [3] proved that . Netanyahu [4] showed that Brannan and Taha [5] introduced certain subclass of the biunivalent functions class . For a brief history and interesting examples of biunivalent functions we refer to [3, 5–11].

The Faber polynomials introduced by Faber [12]. Gong [13] and Schiffer [14] demonstrated the significance of the Faber polynomials in mathematical sciences, especially in geometric function theory. In the literature, there are only a few works determining the general coefficient bounds for the analytic biunivalent functions given by (1) using Faber polynomial expansions. A very little is known about the bounds of Maclaurin’s series coefficient for by using a Faber polynomials. For more study about Faber polynomials we refer to [15–21].

Using the technique of convolution, Ruscheweyh [22] defined the operator on the class of analytic functions as For , we obtain The expression is called an th-order Ruscheweyh derivative of and the symbol stands for Hadamard product (or convolution).

For and , , the number is defined in [23] as For any nonnegative integer the -number shift factorial is defined as We have . Throughout in this paper we will assume to be fixed number between and .

The -derivative operator or -difference operator for is defined as It can easily be seen that for and The -generalized Pochhammer symbol for and is defined as and, for , let -gamma function be defined as For Ruscheweyh -differential operator was defined by Aldweby and Darus [24] (see also [23]), as If , equality (17) implies which is the well known recurrent formula for Ruscheweyh differential operator.

in the present paper we introduce new subclass of the function class , involving Ruscheweyh -differential operator By using Faber polynomial coefficient techniques we determine estimates for the general coefficient bounds for and also estimates on the coefficients and for functions in the new subclass of function class . Several related classes are also considered, and connections to earlier known results are also defined.

Definition 1. A function , , , , and ; we introduce a new class of biunivalent functions as if and only if whereand is defined by (3).
On specializing the parameters , , and , one can state the various new subclasses as illustrated in the following definition.

Definition 2. For , , , and A function, , is in the class if the following conditions are satisfied: where , are given by (20), and is defined by (3).

Definition 3. For , , , and A function, , is in the class if the following conditions are satisfied: where , are given by (20) and is defined by (3).

Definition 4. For , and A function, , is in the class if the following conditions are satisfied: where , are given by (20) and is defined by (3).
It is well known thatSpecial Cases (i)For and , the class ; see [21].(ii)For , , and , the class ; see [25].(iii)For , , , and , the class ; see [2].(iv)For , , and , the class ; see [26].

Lemma 5 (see [27]). Let the Schwarz function be given by then

2. Main Results

Using the Faber polynomial expansion of functions of the form (1), the coefficients of its inverse map may be expressed as [16] given by wheresuch that with is a homogeneous polynomial in the variables ; see [28]. In particular, the first three terms of are in general, for any and , an expansion of (for details we refer [16, 29]), which is as follows: where and, by [28], while , and the sum is taken over all nonnegative integer satisfying Evidently, [15], or, equivalently, while , and the sum is taken over all nonnegative integer satisfying It is clear that ; the first and last polynomials are ,

Theorem 6. For , , , and If , if , , then

Proof. For the function of the form (1), we haveand, for its inverse map , we havewhere is given by (16) and
Since both function and its inverse map are in , by the definition of subordination there exist two Schwarz functions and , where , We have wherewhereIn general [15, 16] for any and , an expansion of ,where is a homogeneous polynomial of degree in the variables
For the coefficients of the Schwarz functions and , and [9].
Comparing the corresponding coefficients of (35) and (37), we have Similarly, corresponding to coefficients of (36) and (39), we haveNote that, for , , we have Taking the absolute values of (44) and (45), we haveBy using , , from (20) and from (16) in (47), we havewhich completes the proof of theorem.

By putting , in Theorem 6, we have the following corollary.

Corollary 7. For , , and , if and if , , thenFor , , in Theorem 6, we have the following corollary.

Corollary 8. For , , if and if , , thenFor , in Theorem 6, we obtain the following corollary.

Corollary 9 (see [21]). For and , , if and if , , thenFor , , and in Theorem 6, we obtain the following corollary.

Corollary 10 (see [25]). For and , if and if , , thenFor , , , in Theorem 6, we obtain the following corollary.

Corollary 11. For , if and if , , then

For , , in Theorem 6, we obtain the following corollary.

Corollary 12 (see [30]). For and , if and if , , then

Theorem 13. For , , , and , if , then

Proof. Replacing by and in (44) and (45), respectively, we have From (56) and (58) we haveUsing (20) and (16) in (61), we haveAdding (57) and (59) we have Taking absolute values of both sides of (63) and applying the estimates , of Lemma 5 and , , we haveusing (20) and (16) in (64), we haveNow, in order to find , we subtract (59) from (57) and we have Using (60) in (66), we haveTaking the modulus of (67), we haveBy using the estimates , , of Lemma 5, and , , in (68), we have and using (56) in (69) we haveUsing (20) and (16) in (70), we haveAgain using (64) in (69) we haveFrom (59) we have Using , of Lemma 5 and (20), (16), on (73), we have

For , in Theorem 13, we obtain the following corollary.

Corollary 14. For , , and , if , then

For and , in Theorem 13, we obtain the following corollary.

Corollary 15. For , , if , then

For , , in Theorem 13, we obtain the following corollary.

Corollary 16 (see [21]). For , , and , if , then

For , , , in Theorem 13, we obtain the following corollary.

Corollary 17 (see [30]). For and , if , then

For , , , , in Theorem 13, we obtain the following [21].

Corollary 18 (see [21]). For , if , then

For , , , in Theorem 13, we obtain the following corollary.

Corollary 19 (see [25]). For , , and , if , then

Disclosure

The present address of Zahid Shareef is Division of Engineering, Higher Colleges of Technology, P.O. Box 4114, Fujairah, UAE.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by MOHE Grant FRGS/1/2016/STG06/UKM/01/1.

References

P. L. Duren, Univalent Functions, Grundehren der Math. Wiss, Springer, New York, NY, USA, 1983.

H. M. Srivastava, A. K. Mishra, and P. Gochhayat, “Certain subclasses of analytic and bi-univalent functions,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 23, no. 10, pp. 1188–1192, 2010.

D. A. Brannan and J. Clunie, “Aspects of contemporary complex analysis,” in Proceedings of the NATO Advanced study Institute Held at University of Durham, Academic Press, New York, NY, USA, 1979.

E. Netanyahu, “The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|<1,” Archive for Rational Mechanics and Analysis, vol. 32, no. 2, pp. 100–112, 1969.

D. A. Brannan and T. S. Taha, “On some classes of bi-univalent functions,” Studia Universitatis Babes-Bolyai Mathematica, vol. 31, no. 2, pp. 70–77, 1986.

S. Altinkaya and S. Yalcin, “Initial coefficient bounds for a general class of biunivalent functions,” International Journal of Analysis, vol. 2014, Article ID 867871, 4 pages, 2014.

B. S. Keerthi and B. Raja, “Coefficient inequality for certain new subclasses analytic bi-univalent functions,” Theory. Math. Appl, vol. 3, no. 1, pp. 1–10, 2013.

M. Lewin, “On a coefficient problem for bi-univalent functions,” Proceedings of the American Mathematical Society, vol. 18, pp. 63–68, 1967.

X.-F. Li and A.-P. Wang, “Two new subclasses of bi-univalent functions,” International Mathematical Forum, vol. 7, pp. 1495–1504, 2012.

N. Magesh and J. Yamini, “Coefficient bounds for certain subclasses of bi-univalent functions,” International Mathematical Forum. Journal for Theory and Applications, vol. 8, no. 25-28, pp. 1337–1344, 2013.

S. Porwal and M. Darus, “On a new subclass of bi-univalent functions,” Journal of the Egyptian Mathematical Society, vol. 21, no. 3, pp. 190–193, 2013.

G. Faber, “Uber polynomische Entwickelungen,” Mathematische Annalen, vol. 57, no. 3, pp. 1569–1573, 1903.

S. Gong, “The Bieberbach conjecture,translated from the 1989 Chinese original and revised by the author,” in AMS/IP Studies in Advanced Mathematics, vol. 12, American Mathematical Society, Providence, RI, USA, 1999.

M. Schiffer, “Surun probleme d’extr emum de la representation conforme,” Bull. Soc. Math. France, vol. 66, pp. 48–55, 1938.

H. Airault, “Remarks on Faber polynomials,” International Mathematical Forum, vol. 3, no. 9, pp. 449–456, 2008.

H. Airault and A. Bouali, “Differential calculus on the Faber polynomials,” Bulletin des Sciences Mathématiques, vol. 130, no. 3, pp. 179–222, 2006.

S. Bulut, “Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions,” Comptes Rendus Mathematique, vol. 352, no. 6, pp. 479–484, 2014.

S. G. Hamidi and J. M. Jahangiri, “Faber polynomial coefficient estimates for analytic bi-close-to-convex functions,” Comptes Rendus Mathematique, vol. 352, no. 1, pp. 17–20, 2014.

S. G. Hamidi, S. Abd Halim, and J. M. Jahangiri, “Faber polynomial coefficient estimates for meromorphic Bi-starlike functions,” International Journal of Mathematics and Mathematical Sciences, vol. 2013, Article ID 498159, 2013.

S. G. Hamidi and J. M. Jahangiri, “Faber polynomial coefficients of bi-subordinate functions,” Comptes Rendus Mathematique, vol. 354, no. 4, pp. 365–370, 2016.

S. Bulut, “Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions,” Filomat, vol. 30, no. 6, pp. 1567–1575, 2016.

S. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109–115, 1975.

S. Kanas and D. Răducanu, “Some class of analytic functions related to conic domains,” Mathematica Slovaca, vol. 64, no. 5, pp. 1183–1196, 2014.

H. Aldweby and M. Darus, “Some subordination results on -analogue of Ruscheweyh differential operator,” Abstract and Applied Analysis, vol. 2014, Article ID 958563, 6 pages, 2014.

H. M. Srivastava, S. S. Eker, and R. M. Ali, “Coefficient bounds for a certain class of analytic and bi-univalent functions,” Filomat, vol. 29, no. 8, pp. 1839–1845, 2015.

B. A. Frasin and M. K. Aouf, “New subclasses of bi-univalent functions,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 24, no. 9, pp. 1569–1573, 2011.

F. R. Keogh and E. P. Merkes, “A coefficient inequality for certain classes of analytic functions,” Proceedings of the American Mathematical Society, vol. 20, pp. 8–12, 1969.

H. Airault, “Symmetric sums associated to the factorizations of Grunsky coefficients,” in Proceedings of the Groups and Symmetries, Montreal, Canada, April 2007.

H. Airault and J. Ren, “An algebra of differential operators and generating functions on the set of univalent functions,” Bulletin des Sciences Mathématiques, vol. 126, no. 5, pp. 343–367, 2002.

J. M. Jahangiri and S. G. Hamidi, “Coefficient estimates for certain classes of bi-univalent functions,” International Journal of Mathematics and Mathematical Sciences, vol. 2013, Article ID 190560, 2013.