Abstract
In this paper, we use the Faber polynomial expansion to obtain bounds for the general coefficients of bi-univalent functions in the family of analytic functions in the open unit disk. Estimation of the bound value for the initial coefficients of the functions in these classes is also established.
1. Introduction
Let denote the class of all functions of the form which are analytic in the open unit disc . Also, let be the class of all functions in which are univalent in . For Airault and Bouali ([1], page 184) used Faber polynomial to show that where is the Faber polynomial defined by
The first terms of the Faber polynomial , , are given by (e.g., see [2], page 52)
The Koebe one-quarter theorem ([3], page 31) ensured that the range of each function of the class contains the disc . Therefore, the univalent function has an inverse which is defined by
The inverse of the function has Taylor expansion given by (see [1], page 185) where the coefficients are given by and is homogeneous polynomial of degree in the variables (see [4], page 349 and [1], pages 183 and 205).
Lemma 1 (Schwarz lemma ([3], page 3). If is analytic in the unit disc with and in then and in .
Let and be analytic functions in ; we say that the function is subordinate to , written as follows: if there exists a Schwarz function , which (by definition) is analytic in with and , such that for all . In particular, if the function in , then we have the following equivalence relation (cf., e.g., [5, 6]; see also [3]):
Let be analytic function with positive real part in , satisfying Also, let be symmetric with respect to the real axis. Such a function has a Taylor series of the form
Attiya [7] introduced the operator , where is defined by with and . Also, when ; Here, is the generalized Mittag-Leffler function defined by [8] (see also [7]), and the symbol denotes the Hadamard product or convolution.
A detailed investigation of the Mittag-Leffler function has been studied by many authors (see, e.g., [8–12]).
Attiya [7] noted that
From (13), it follows (see [7])
Also, note the following: (1)(2)(3)(4)(5)(6)
Definition 2. A function is said to be in the class if it satisfies where is defined by (11).
Definition 3. A function is said to be in the class if it satisfies where is defined by (11).
With the virtue of (14) and (15), we find that and when is the well-known class of starlike function of order .
A single-valued function analytic in a domain is said to be univalent, if it never takes the same value twice in , that is, if for all points and in with (see [3], page 26). Also, a function is said to be bi-univalent in if and its inverse map are univalent in .
We denote by the class of bi-univalent functions in given by (1). The class of analytic bi-univalent functions was introduced and studied by Lewin [13] and showed that . Recently, many authors found nonsharp estimates on the first two Taylor–Maclaurin’s coefficients and .
The following are examples of bi-univalent functions in :
For various subclasses of bi-univalent functions, see, for example, [14–26].
Definition 4. A function given by (1) is said to be in the class if both and its inverse map are in . Also, a function given by (1) is said to be in the class if both and its inverse map are in .
In this paper, we use the Faber polynomial expansion to obtain bounds for the general coefficients of bi-univalent functions in and as well as we estimate the bounds of the initial coefficients of the functions in these classes.
Unless otherwise mentioned, we assume throughout this paper that and . Also, when ; .
2. Coefficient Estimates of and
Theorem 5. Let the function given by (1) be in the class . Also, let and for , where is a divisor of . Then, where is defined in (11).
Proof. If we set then with ; using relation (14), we have
Since and its inverse are in , , therefore, there are analytic functions and with , and , such that
where ; then, , with ; using (7), we have .
The functions and are defined by
It is well known that (see Duren [3], page 265)
Also, we have
In general (see [21], page 649), the coefficients are given by
where is a homogeneous polynomial of degree in the variables .
Using the Faber polynomial expansion, (2) yields the following identities:
Comparing the corresponding coefficients of (27) and (24) yields
and similarly, from (28) and (25), we have
Since and for , and is a divisor of then by using (29) and (30), we have
also, under the condition and for and using the relation between and , we have . Then, (31) yields
Using either (31) or (33), we have substitute of . This completes the proof of the theorem.
Putting in Theorem 5, we have the following:
Corollary 6. Let the function given by (1) be in the class ; if for , then where is defined in (11).
Using the same technique used in Theorem 5, we get the following theorem:
Theorem 7. Let the function given by (1) be in the class Also, let and for , where is a divisor of Then, where is defined in (11).
Putting in Theorem 7, we have the following:
Corollary 8. Let the function given by (1) be in the class ; if for , then where is defined in (11).
To prove our next theorem, we shall need the following lemma.
Lemma 9 [3, 21]. Let the function be a Schwarz function with , . Then for ,
Theorem 10. Let the function given by (1) be in the class Then, where and .
Proof. Putting and in (29) and (30), respectively, we find that
Moreover, we have and .
Therefore, (41) and (42) imply
Apply Lemma 9.
Case I: if , then both and have maximum value at 1 when . Also, and have maximum value at when , where and
Case II: if , then both and have maximum value at when and and have maximum value at when . Then, we have (38)
Moreover, from (41) and (42), we get
It follows from (43) and (44) that
Applying the cases mentioned above for and , we have (39). This completes the proof of the theorem.
Using the same technique used in Theorem 10, we get the following theorem:
Theorem 11. Let the function given by (1) be in the class . Then, where and .
3. Conclusion
By using the Faber polynomial expansion, we obtain bounds for the general coefficients of bi-univalent functions for functions in the classes and in ; also, estimation of the bound value for the initial coefficients of the functions in the classes and is established.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
The authors contributed equally to the writing of this paper. All authors approved the final version of the manuscript.
Acknowledgments
This research has been funded by the Scientific Research Deanship at University of Ha'il, Saudi Arabia, through project number RG-21050.