Research Article | Open Access

# Certain Subclasses of Bi-Close-to-Convex Functions Associated with Quasi-Subordination

**Academic Editor:**Luigi Rodino

#### Abstract

In the present investigation, we introduce certain new subclasses of the class of biunivalent functions in the open unit disc defined by quasi-subordination. We obtained estimates on the initial coefficients and for the functions in these subclasses. The results present in this paper would generalize and improve those in related works of several earlier authors.

#### 1. Introduction and Preliminaries

Let be the class of functions of the formwhich are analytic in the open unit disc . Further, let be the class of functions and univalent in .

By , we denote the class of bounded or Schwarz functions satisfying and which are analytic in the unit disc and of the form

Firstly, it is necessary to recall some fundamental definitions to acquaint with the main content:

The functions in the class are invertible but their inverse function may not be defined on the entire unit disc . The Koebe-one-quarter theorem [1] ensures that the image of under every function contains a disc of radius . Thus every univalent function has an inverse , defined bywhere A function is said to be biunivalent in if both and are univalent in U.

Accordingly, a function is said to be bistarlike, biconvex, bi-close-to-convex, or bi-quasi-convex if both and are starlike, convex, close-to-convex, or quasi-convex respectively.

Let denote the class of biunivalent functions in given by (1). Examples of functions in the class areand so on. However, the familiar Koebe function is not a member of .

Let and be two analytic functions in . Then is said to be subordinate to (symbolically ) if there exists a bounded function such that . This result is known as principle of subordination.

Robertson [2] introduced the concept of quasi-subordination in 1970. For two analytic functions and , the function is said to be quasi-subordinate to (symbolically ) if there exist analytic functions and with , and such thator equivalently Particularly if , then , so that in . So it is obvious that the quasi-subordination is a generalization of the usual subordination. The work on quasi-subordination is quite extensive which includes some recent investigations [3–6].

Lewin [7] investigated the class of biunivalent functions and obtained the bound for the second coefficient. Brannan and Taha [8] considered certain subclasses of biunivalent functions, similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike, and convex functions. They introduced bistarlike functions and biconvex functions and obtained estimates on the initial coefficients. Also the subclasses of bi-close-to-convex functions were studied by various authors [9–11].

Motivated by earlier work on bi-close-to-convex and quasi-subordination, we define the following subclasses.

Also it is assumed that is analytic in with and let

*Definition 1. *For , a function given by (1) is said to be in the class if there exists a bistarlike function such thatwhere , , and .

For , the class reduces to , the class of bi-close-to-convex functions of complex order defined by quasi-subordination.

*Definition 2. *For , a function given by (1) is said to be in the class if there exists a biconvex function and satisfy the following conditions:where , , and .

It is interesting to note that, for , is the subclass of bi-close-to-convex functions of complex order defined by quasi-subordination. Also for , is the class of bi-quasi-convex functions of complex order defined by quasi-subordination.

For deriving our main results, we need the following lemmas.

Lemma 3 (see [12]). *If is family of all functions analytic in for which and have the form for , then for each .*

Lemma 4 (see [13]). *If is a starlike function, then*

Lemma 5 (see [13]). *If is a convex function, thenAlong with the above lemmas, the following well known results are very useful to derive our main results.**Let be an analytic function in of the form (1), then , if is starlike and , if is convex.*

#### 2. Coefficient Bounds for the Function Class

Theorem 6. *If , then*

*Proof. *As , so by Definition 1 and using the concept of quasi-subordination, there exist Schwarz functions and and analytic function such thatwhere and .

Define the functions and byUsing (21) and (22) in (19) and (20), respectively, it yieldsButAgain using (9) and (10) in (21) and (22), respectively, we getUsing (25) and (27) in (23) and equating the coefficients of and , we getAgain using (26) and (28) in (24) and equating the coefficients of and , we getFrom (29) and (31), it is clear thatTherefore on applying triangle inequality and using Lemma 3, (34) yields As is starlike, so it is well known that , (35) gives Adding (30) and (32), it yields Using (36) and on applying triangle inequality in (37), we obtain As is starlike, so using in (38), it yieldsSo, result (17) can be easily obtained from (36) and (39).

Now subtracting (32) from (30), we obtain Applying triangle inequality and using Lemma 3 in (40), it yields Again adding (30) and (32) and applying triangle inequality, we getUsing (42) in (41), it givesOn applying Lemma 4 in (43), the result (18) is obvious.

For , Theorem 6 gives the following result.

Corollary 7. *If , then*

#### 3. Coefficient Bounds for the Function Class

Theorem 8. *If , then*

*Proof. *On applying Lemmas 3 and 5 and following the arguments as in Theorem 6, the proof of this theorem is obvious.

On putting , Theorem 8 gives the following result.

Corollary 9. *If , then**On putting , Theorem 8 gives the following result.*

Corollary 10. *If , then*

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### References

- P. L. Duren,
*Univalent Functions*, Springer, New York, NY, USA, 1983. View at: MathSciNet - M. S. Robertson, “Quasi-subordination and coefficient conjecture,”
*Bulletin of the American Mathematical Society*, vol. 76, pp. 1–9, 1970. View at: Publisher Site | Google Scholar | MathSciNet - O. Altintas and S. Owa, “Mazorizations and quasi-subordinations for certain analytic functions,”
*Proceedings of the Japan Academy, Series A, Mathematical Sciences*, vol. 68, no. 7, pp. 181–185, 1992. View at: Publisher Site | Google Scholar | MathSciNet - S. Y. Lee, “Quasi-subordinate functions and coefficient conjectures,”
*Journal of the Korean Mathematical Society*, vol. 12, no. 1, pp. 43–50, 1975. View at: Google Scholar | MathSciNet - N. Magesh, V. K. Balaji, and J. Yamini, “Certain subclasses of bistarlike and biconvex functions based on quasi-subordination,”
*Abstract and Applied Analysis*, vol. 2016, Article ID 3102960, 6 pages, 2016. View at: Publisher Site | Google Scholar | MathSciNet - F. Y. Ren, S. Owa, and S. Fukui, “Some inequalities on quasi-subordinate functions,”
*Bulletin of the Australian Mathematical Society*, vol. 43, no. 2, pp. 317–324, 1991. View at: Publisher Site | Google Scholar | 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 Site | Google Scholar | MathSciNet - D. A. Brannan and T. S. Taha, “On some classes of bi-univalent functions,” in
*Mathematical Analysis and Its Applications*, S. M. Mazhar, A. Hamoni, and N. S. Faour, Eds., vol. 3 of*KFAS Proceedings Series*, pp. 53–60, Pergamon Press, Oxford, UK, 1985, See also Studia Univ. Babes-Bolyai Math., 31 (1986), No. 2, 70-77. View at: Google Scholar - F. M. Sakar and H. O. Guney, “Coefficient bounds for a new subclass of analytic bi-close-to-convex functions by making use of Faber polynomial expansion,”
*Turkish Journal of Mathematics*, vol. 41, no. 4, pp. 888–895, 2017. View at: Publisher Site | Google Scholar | MathSciNet - B. Seker and S. Sumer Seker, “On subclasses of bi-close-to-convex functions related to the odd-starlike functions,”
*Palestine Journal of Mathematics*, vol. 6, pp. 215–221, 2017. View at: Google Scholar | MathSciNet - C. Selvaraj and T. R. K. Kumar, “Bi-univalent coefficient estimates for certain subclasses of close-to-convex functions,”
*International Journal of Mathematics and Its Applications*, vol. 3, no. 4-D, pp. 69–74, 2015. View at: Google Scholar - C. Pommerenke,
*Univalent Functions*, Vandenhoeck & Ruprecht, Göttingen, Germany, 1975. View at: MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2019 Gurmeet Singh 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.