Abstract

In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. This formula is applied to a certain class of bi-univalent functions and solve the -th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its -th term.

1. Introduction

The structural properties and information about Geometric Function Theory depends on the estimation of coefficient of analytic functions. For example, the second coefficient estimation in the set of univalent functions gives the growth, distortion bounds, and also covering theorems. The study of coefficient bounds for the classes of bi-univalent functions was first investigated by Levin [1] in 1967, but the interest sparks among the researchers when Brannan and Taha [2] conjectured the coefficient bounds for the classes of bi-univalent functions. A series of coefficients investigation have been carried out recently in [311] and coefficient properties by Rehman et al. [12]. In fact, the work of Srivastava et al. [13] has made a huge impact on the development of bi-univalent functions and appeared frequently in the literature ever since the publication of their pioneering work. In a recent development, Srivastava et al. [14] have made the use of Faber polynomial expansions with -analysis to determine the bounds for the -th coefficient in the Taylor–Maclaurin series expansions. In addition to this, Srivastava et al. [15] made the use of a linear combination of three functions (, and ) with the technique involving the Faber polynomials and determined the coefficient estimates for the general Taylor–Maclaurin functions belonging to the bi-univalent function.

Let denotes the class of analytic functions in the unit disk, of the following form:

Furthermore, we represent by the class of univalent functions, which is defined as a function is called univalent on (or schlicht or one-to-one) if for all with . The leading member of this class is the famous Köebe’s function of the form . Some other examples are , , and (see [16]). Next, we denote by the class of bi-univalent functions that states the class of functions is said to be bi-univalent in the unit disk if both and are univalent in . For example, , , and (for details, see [10,13]). According to Köebe’s One Quarter Theorem (see [17], p. 31), the range of every function of class contains the disk . Köebe’s theorem ensures that the image of a unit disc , under every univalent function , contains a disk of radius . Therefore, every univalent function in has an inverse defined by

These type inverse functions can easily be verified bywhere .

Let , , and be analytic functions in the open unit disc . The function is said to be the subordinate to , expressed as , if there exists a Schwartz function , that is, , , and . Particularly, if the function is univalent in the unit disc , then if and only if and (cf. [18, 19]).

2. Discussion

Löwner [20] and Pommerenke [19] proved that the inverse of Köebe’s function concedes the best bounds for all . However, new techniques have been adopted recently to determine the peculiar behavior of coefficients for various subclasses of (see, for example, [2124]).

The series expansion of the inverse of in some disk about the origin is given by

Moreover, a function which is univalent in a neighborhood about the origin with its inverse satisfies the condition , and then, equation (4) can be written aswhere is the Faber polynomial expansion of functions of the form (1), wherewith such that and is a homogeneous polynomial in variables . For more details, one may refer to the expansion of , and for the coefficients of its inverse functions, see Theorem 6.1 in [25] (p. 209) and [6]. Likewise, using the general term , we can compute the first six terms as follows:

The Faber polynomials introduced in 1903 by Faber [26] (also see [27]) play an important role in various areas of mathematical sciences, in particular the geometric function theory (Gong [28], Chapter III, and Schiffer [29]). The calculus of Faber polynomials gets more importance, especially when it was found useful in the study of inverse functions (see for details [25, 3032]). Based on the implication of Faber polynomial expansions in determining the coefficient estimations of the bi-univalent functions and following the work of [1, 3, 9, 10, 1315, 33, 34] and [35], we are motivated to derive new type of polynomials that collaborate with the Faber polynomial expansion to estimate the coefficient bounds for a certain class of bi-univalent functions beyond .

Throughout the article, we consider to be analytic with its positive real part on the unit disk ; obeying the conditions , , and is symmetric with respect to real axis. This type of function can be expressed as series expansion of the following form:for all with .

We now define the class with the following conditions.

Definition 1. Let , , and . A function is in the class if the following subordinations hold:where .

Remark 1. If we set in , the class interacts with the class introduced in [3]. Next, if we set , and in Definition 1, we get the following class .

Definition 2. (see [3]). A function is in the class , if the following subordinations hold:In fact, the class we introduced is mainly inspired by Bansal and Sokol [3], Kumar et al. [7], and [13, 33]. For example, for in , the class interacts with [7]; for in , the class meets with [3]; and for , , (), the class joins to the function class given in [33].
In order to prove our main result, we need the following lemma.

Lemma 1. (see [17]). Let the function be given by the seriesthen the sharp estimate , holds.
The two important functions that frequently appear in the literature of bi-univalent functions are considered by many authors for determining the initial coefficient bounds of certain class of bi-univalent functions. Functions and , respectively, are defined bywhere by Lemma 1, .

3. Preliminary Results

In order to accomplish the intended formula, we introduce two polynomials and define them as and polynomials, respectively, as follows:where in equation (15),and in equation (16),

The triangular arrays in (17) and (18) lead us to form the general term of and , respectively, as follows:

Therefore, equation (15) becomesand equation (16) yields :

Now, using the above series of expressions in (17) and (18), we enlist below some of the terms such as , , , , , , etc. and , , , , , , etc., respectively:and these expressions go on.

Upon using equation (18), we obtain the following list of :and the expressions go on.

Note 1. The following notations will be used in coming sections:(1): the set of all possible for (2): list of usable (3): list of all possible , where is chosen from (8)(4): number of previous for (5): , shows the degree of each combination of and , represents the number of , in a single combination(6): the multipliers of in terms of .

Remark 2. The relation to determine , associated with , is given bywhere is taken from the coefficient of . For example, referring to (6) in Note 1, if we need to find for the coefficient of , then . This implies . Therefore, for the coefficient of , the multipliers of .

4. Main Results

The study of bi-univalent functions shows that most authors use the comparison method between their newly introduced bi-univalent classes and the analytic functions under certain conditions to estimate the coefficient bounds for their function classes. The same type of studies can be found in [3, 7, 10] and many others. The main crux to estimate the -th coefficient bounds of such bi-univalent functions lies in the generalization of the two analytic functions defined in (13) and (14) and the generalization of the class of certain bi-univalent functions. Hence, we present the generalization of two analytic functions that assist in structuring the formula for estimating the -th coefficient bound for certain class of bi-univalent functions.

4.1. Analytic Functions Correlative to Bi-Univalent Functions

Theorem 1. If and the function is analytic in the unit disk, then for a certain bi-univalent function, the comparative coefficients of may be represented by the following expansions:

Proof. The given function is analytic in as , and thus, function has the following Taylor–Maclaurin series expansion:Since , the analytic function has a positive real part on , and in view of Lemma 1, we have . Hence, solving (27) for , we getNow, utilizing equations (8) and (9) and equations (27) and (28), we obtainSince a function belonging to the class of bi-univalent functions is in the form of (1), a calculation shows that the has similar expansion as in expression (3). So, by using (1) in (29), we obtain

Remark 3. In the above equation, the left-hand side is the generalized form of the class , while the right-hand side is the newly inducted polynomial function associated with (8), (27), and (28).

Remark 4. The polynomial on the right-hand side of (30) is extendable to and thus needs to determine the coefficient of .
Now, by comparing the coefficients of , we have the following expressions:Thus, by iteration, we deduce a general formula that relates (8) and (9) with polynomial defined in (15). So the acquired formula that finds the series of expressions asserted in (31) is given as follows:This proves our assertion in (26).

Note 2. Observe that the above equation consists of two parts in which the first part of takes single values, while the second part takes simultaneously two different values such that for . In the case of , the whole term is omitted. Also, shows the degree of each combination of such that the values of are chosen from expression (17) or from (19). For numerical explanation of (32), see Example 1 in Section 4.

Theorem 2. If and the function is analytic in the unit disk, then for a certain bi-univalent function, the comparative coefficients of may be represented by the following expansions:

Proof. The given function is analytic in as , and thus, function has the following Taylor–Maclaurin series expansion:Since , the analytic function has a positive real part on , and in the view of Lemma 1, we have . Thus, solving (34) for , we obtainso, from equations (8) and (10) and equations (34) and (35), we getSince a function belonging to the class of bi-univalent functions has the Maclaurin series given by (1), a calculation shows that the inverse of has similar expansion as in (3).
Now, by using (2), (5), and (7) in (36), we getThus, by comparing the coefficients of , we haveSimilar to (32), we obtain the following formula that relates (8) and (10) with polynomial defined in (16):where represents the degree of each combination of such that the values of are chosen from expression (18) and values of are obtained from (7) or in general from (6). This statement completes the proof.

4.2. Applicability of and Polynomials

In this section, we employ and polynomials on our class stated in Definition 2. Here, the aim is to test our newly constructed relation defined in (30) on our class in order to obtain the initial coefficient estimation.

Theorem 3. Let be of the form (2), then

Proof. Let and , then there exist analytic functions with , satisfyingApproaching towards the estimation of , we compare the coefficients of . For this purpose, utilizing the formula in (32) by putting , we get

Note 3. In the case of singleton , the second part of the equation (32) is no more usable as it takes simultaneously two different values (see Note 1).
Therefore, by putting in (32), we obtainFor the next corresponding expression, utilizing the formula in (39) by putting , we getOnce again in view of Note 3, the second part of the formula in (39) is dropped as it needs simultaneously two values (see Note 1). Thus, putting in (39), we obtainsuch that the value of is provided by the Faber polynomial expansions given in (7) or could be calculated from (6).

Remark 5. Note that, by comparing (44), (46), and (7) together with (23) and (24), we get that leads to .
Now, by squaring (44), we obtain the following initial value of :Again, for the conclusive value of , we compare the coefficients of and using the formula in (32) by putting , we getAgain, for a singleton , the second part of the formula is no more valid (see Note 3). Thus, equation (32) reduces toFor the next expression, utilizing the formula in (39) by putting , we getwhere the value of is given by the Faber polynomial expansion defined in (7) or could be calculated from (6).
Now, adding (49) and (50) and then using (17), (18), (23), and (24) together with the value of from (47), we conclude the following value of :Next, subtracting (49) from (50) and then using (17), (18), (23), and (24) together with the value of from (47), we obtain the following value of :So, applying Lemma 1 on (51) and (52), we obtain the proof asserted in (40) and (41).

Remark 6. If we set out the value of , we receive Theorem 1 of [3].

Remark 7. If we put the value of , , and , we receive Theorem 1 of [33].

4.3. Certain Coefficient of

In this subsection, we offer an example that demonstrates the calculation of the certain coefficient of , and here, we take .

Example 1. How to determine the expression by comparing the coefficient of , with the corresponding class of bi-univalent function?
Manipulating (31) for , two extreme terms are and , so we only work for finding the mean terms that is and (also refer to Remark 1):The expression in (53) means that involves only , and , in its final estimation. So, on further exploration, we obtain the following expressions:Note that some of the terms in above equation violates the conditions stated in Theorem 1. Hence, by neglecting those terms, we get