Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor Darul Ehsan, Malaysia Center for Language and Foundation Studies, A’ Sharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Oman Department of Basic Science, College of Health and Applied Sciences, A’ Sharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Oman


Introduction e structural properties and information about Geometric
Function eory depends on the estimation of coefficient of analytic functions. For example, the second coefficient estimation (|a 2 |) in the set of univalent functions gives the growth, distortion bounds, and also covering theorems. e 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 [3][4][5][6][7][8][9][10][11] 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 q-analysis to determine the bounds for the n-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 ((f(z)/z), f′(z), and zf ″ (z)) 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 A denotes the class of analytic functions in the unit disk, U � z ∈ C: |z| < 1 { } of the following form: Furthermore, we represent by T the class of univalent functions, which is defined as a function f ∈ A is called univalent on U (or schlicht or one-to-one) if f(z 1 ) ≠ f(z 2 ) for all z 1 , z 2 ∈ U with z 1 ≠ z 2 . e leading member of this class is the famous Köebe's function of the form K(z) � (z/(1 − z) 2 ). Some other examples are v(z) � z, t(z) � (z/1 − z), and u(z) � (z/1 − z 2 ) (see [16]). Next, we denote by Σ the class of bi-univalent functions that states the class of functions f ∈ A is said to be bi-univalent in the unit disk U if both f(z) and f − 1 (z) are univalent in U. For example, f(z) � (z/1 − z), d(z) � − log(1 − z), and j(z) � (1/2)log(1 + z/1 − z) (for details, see [10,13]). According to Köebe's One Quarter eorem (see [17], p. 31), the range of every function of class T contains the disk w: |w| < (1/4) { }. Köebe's theorem ensures that the image of a unit disc U, under every univalent function f ∈ T, contains a disk of radius (1/4). erefore, every univalent function in f ∈ A has an inverse f − 1 defined by ese type inverse functions can easily be verified by where , and w(z) be analytic functions in the open unit disc U. e function f(z) is said to be the subordinate to g(z), expressed as f(z)≺g(z), if there exists a Schwartz function w, that is, [18,19]).

Discussion
Löwner [20] and Pommerenke [19] proved that the inverse of Köebe's function concedes the best bounds for all |a n |. However, new techniques have been adopted recently to determine the peculiar behavior of coefficients a n for various subclasses of T (see, for example, [21][22][23][24]). e series expansion of the inverse of f ∈ T in some disk about the origin is given by Moreover, a function f(z) which is univalent in a neighborhood about the origin with its inverse satisfies the condition f(f − 1 (w)) � w, and then, equation (4) can be written as . , a n )w n is the Faber polynomial expansion of functions f ∈ A of the form (1), where ) and S j (7 ≤ j ≤ n) is a homogeneous polynomial in variables (a 2 , a 3 , . . . , a n ). For more details, one may refer to the expansion of K − n n− 1 , and for the coefficients of its inverse functions, see eorem 6.1 in [25] (p. 209) and [6]. Likewise, using the general term K − n n− 1 , we can compute the first six terms as follows: e 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]). e calculus of Faber polynomials gets more importance, especially when it was found useful in the study of inverse functions (f − 1 ) (see for details [25,[30][31][32]). 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, 13-15, 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 |a n | ≥ 2: n ∈ N ∘ � 2, 3, 4, . . . { }. roughout the article, we consider G(z) to be analytic with its positive real part on the unit disk U; obeying the conditions G(0) � 1, G ′ (0) > 0, and G(U) is symmetric with respect to real axis. is type of function can be expressed as series expansion of the following form:

Journal of Mathematics
We now define the class ΣH(α, τ, c)(G) with the following conditions.
In order to prove our main result, we need the following lemma.
Lemma 1 (see [17]). Let the function b ∈ T be given by the series then the sharp estimate |h n | ≤ 2, (n ∈ N), holds. e 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 b 1 (z) and b 2 (w), respectively, are defined by where by Lemma 1, |q n | and |v n | ≤ 2.

Preliminary Results
In order to accomplish the intended formula, we introduce two polynomials and define them as R− and X− polynomials, respectively, as follows: where in equation (15), and in equation (16), e triangular arrays in (17) and (18) lead us to form the general term of R n and X n , respectively, as follows: erefore, equation (15) becomes and equation (16) yields t(w): Now, using the above series of expressions in (17) and (18), we enlist below some of the terms such as R 1 , R 2 , R 3 , R 4 , R 5 , R 6 , etc. and X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , etc., respectively: and these expressions go on. Upon using equation (18), we obtain the following list of X n : n ∈ N � 1, 2, 3, . . . { }: and the expressions go on.
Note 1. e following notations will be used in coming sections: (1) n R t i : the set of all possible R ′ s for t i � 1, 2, 3, . . .
: m, shows the degree of each combination of u R t i and t k , represents the number of u R t i , in a single combination (6) B t k u R t i : the multipliers of B t k in terms of u R t i .

Remark 2.
e relation to determine u R t i , associated with B t k , is given by where n is taken from the coefficient of z n : n � 1, 2, 3, . . . { }. For example, referring to (6) in Note 1, if we need to find erefore, for the coefficient of z 4 , the multipliers of

Main Results
e 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. e same type of studies can be found in [3,7,10] and many others. e main crux to estimate the n-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 biunivalent functions. Hence, we present the generalization of two analytic functions that assist in structuring the formula for estimating the n-th coefficient bound for certain class of bi-univalent functions.

Analytic Functions Correlative to Bi-Univalent Functions
is analytic in the unit disk, then for a certain bi-univalent function, the comparative coefficients of z n : n ∈ N, z ∈ C { } may be represented by the following expansions: Proof. e given function b 1 (z) � (1 + s(z)/1 − s(z)) is analytic in U as b 1 (0) � 1, and thus, function b 1 has the following Taylor-Maclaurin series expansion: Since s: U ⟶ U, the analytic function b 1 has a positive real part on U, and in view of Lemma 1, we have |q n | ≤ 2: n ∈ N. Hence, solving (27) for s(z), we get z + q 2 z 2 + · · · + q n z n 2 + q 1 z + q 2 z 2 + · · · + q n z n � R 1 z + R 2 z 2 + · · · + R n z n .
Now, utilizing equations (8) and (9) and equations (27) and (28), we obtain Since a function f belonging to the class of bi-univalent functions is in the form of (1), a calculation shows that the f − 1 (z) � Q(w) 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 ΣH(α, τ, c)(G), while the right-hand side is the newly inducted polynomial function associated with (8), (27), and (28).

Remark 4.
e polynomial on the right-hand side of (30) is extendable to z n and thus needs to determine the coefficient of z n . Now, by comparing the coefficients of z, z 2 , z 3 , . . . , z n , we have the following expressions: us, by iteration, we deduce a general formula that relates (8) and (9) with R− polynomial defined in (15). So the acquired formula that finds the series of expressions asserted in (31) is given as follows: is proves our assertion in (26).
Note 2. Observe that the above equation consists of two parts in which the first part of t i takes single values, while the second part takes simultaneously two different values such that t i ≠ t j for i ≠ j. In the case of t i , t j , t k , (t k − n) < 0, the whole term is omitted. Also, m ∈ N shows the degree of each combination of ( u R t i : t i � 1, 2, 3, . . . { }) such that the values of R n are chosen from expression (17) or from (19). For numerical explanation of (32), see Example 1 in Section 4.

Theorem 2.
If t i , t j , t k , (t k − n) > 0 and the function b 2 (w) � (1 + t(w)/1 − t(w)) is analytic in the unit disk, then for a certain bi-univalent function, the comparative coefficients of w n : n ∈ N, w ∈ C { } may be represented by the following expansions: Proof. e given function b 2 (w) � (1 + t(w)/1 − t(w)) is analytic in U as b 2 (0) � 1, and thus, function b 2 (w) has the following Taylor-Maclaurin series expansion: Since t: U ⟶ U, the analytic function b 2 has a positive real part on U, and in the view of Lemma 1, we have |v n | ≤ 2: n ∈ N. us, solving (34) for t(w), we obtain � v 1 w + v 2 w 2 + · · · + v n w n 2 + v 1 w + v 2 w 2 + · · · + v n w n � X 1 w + X 2 w 2 + · · · + X n w n , Journal of Mathematics so, from equations (8) and (10) and equations (34) and (35), we get Since a function f belonging to the class of bi-univalent functions has the Maclaurin series given by (1), a calculation shows that the inverse of Q(w) � f − 1 (w) has similar expansion as in (3). Now, by using (2), (5), and (7) in (36), we get (37) us, by comparing the coefficients of w, w 2 , w 3 , . . . , w n , we have such that the value of (K − 2 1 /2) is provided by the Faber polynomial expansions given in (7) or could be calculated from (6).

Conclusions
We presented an idea to generalize the analytic functions that are correlative to certain bi-univalent functions. By utilizing the R− and X− polynomials, we derived the initial estimates for our newly defined class of a bi-univalent function ΣH(α, τ, c)(G) (see Definition 1 and eorem 3). Furthermore, we provided an example to calculate the coefficients of z n in terms of R ′ s (see Example 1). In addition to this, we have given the n-th term of several classes (see (56), (59), and (61)). e R− and X− coefficients defined in (32) and (39) allow researchers to compare the n-th coefficients of (z) or (w) with Faber polynomial expansions to explore the n-th coefficient estimate for a certain biunivalent function.

Data Availability
All data generated or analysed during the study are included within the submitted article.

Disclosure
All the authors agreed with the content of the manuscript.

Conflicts of Interest
e authors declare that there are no conflicts of interest.