Abstract and Applied Analysis
Volume 2019 (2019), Article ID 6947020, 6 pages
https://doi.org/10.1155/2019/6947020

Research Article

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

Gurmeet Singh,¹ Gurcharanjit Singh,² and Gagandeep Singh³

¹Principal, Patel Memorial National College, Rajpura, Punjab, India
²Research Scholar, Department of Mathematics, Punjabi University, Patiala, Punjab, India
³Department of Mathematics, Majha College for Women, Tarn Taran Sahib, Punjab, India

Correspondence should be addressed to Gagandeep Singh; kamboj.gagandeep@yahoo.in

Received 13 January 2019; Accepted 7 March 2019; Published 1 April 2019

Academic Editor: Luigi Rodino

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.

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 form

which 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 by

where

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 are

and 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 that

or 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 that

where , , 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, then

Along 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 that

where and .
Define the functions and by

Using (21) and (22) in (19) and (20), respectively, it yields

But

Again using (9) and (10) in (21) and (22), respectively, we get

Using (25) and (27) in (23) and equating the coefficients of and , we get

Again using (26) and (28) in (24) and equating the coefficients of and , we get

From (29) and (31), it is clear that

Therefore 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 yields

So, 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 get

Using (42) in (41), it gives

On 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.

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