A New Method for Estimating General Coefficients to Classes of Bi-univalent Functions

This study establishes a new method to investigate bounds of a k ; k ≥ n , for certain general classes of bi-univalent functions. The results include a number of improvements and generalizations for well-known estimations. We also discuss bounds of na 2 n − a 2 n − 1 and consider several corollaries, remarks, and consequences of the results presented in this paper


Introduction and Preliminary
In the usual notation, let A denote the class of functions f ζ in the form which are analytic in the open unit disk U = ζ ∈ ℂ ζ < 1 and normalized by f 0 = f ′ 0 − 1 = 0. We also denote by S the subclass of A consisting of univalent (one-to-one) functions in U. Further, the class P is consisting of analytic functions p ζ = 1 + ∑ ∞ n=1 c n ζ n satisfying p 0 = 1 and Re p ζ > 0, ζ ∈ U .Note that c n ≤ 2, for n ≥ 1, by the Carathéodory lemma.It is well known that every function f ∈ S, in the form (1), has an inverse function f −1 defined by f −1 f ζ = ζ, ζ ∈ U and f f −1 w = w, w < 1/4 , according to the Koebe one-quarter theorem (see [1]).In fact, the inverse function g = f −1 is given by g w = f −1 w = w − a 2 w 2 + 2a 2  2 − a 3 w 3 − 5a 3  2 − 5a 2 a 3 + a 4 w 4 +⋯ A function f ∈ A is said to be bi-univalent in U if both f and f −1 are univalent in U.The class of bi-univalent functions in U is denoted by Σ.As can be observed from studies [2][3][4], research on the class of bi-univalent functions started some time ago, probably around the year 1967.Indeed, the bounds for the coefficients of functions in Σ were first investigated by Lewin [2], where he showed that a 2 < 1 51.Later, Brannan and Clunie [4] conjectured that a 2 ≤ 2, and Netanyahu [3] proved that a 2 ≤ 4/3 for functions of Σ whose images cover the open unit disk.The best known estimate for a 2 is 1.485 given by Tan [5].Brannan and Taha [6] have introduced nonsharp estimates for the first two coefficients of the strongly bistarlike, bistarlike, and biconvex function classes.In the recent years, interest in the subject has returned; for example, numerous publications have been published since 2010 .
Since the condition of bi-univalency renders the behavior of the higher coefficients unpredictable, determining the boundaries for a n ; n ≥ 4 is a notable problem in geometric function theory.Ali et al. [28] also proclaimed it to be an open topic.Accordingly, numerous writers have estimated the initial nonzero coefficient a n for a variety of Σ subclasses using the Faber polynomials (see [29][30][31][32][33][34][35][36]).Al-Refai and Ali, on the other hand, recently developed an alternative approach to estimating a n .They have in fact validated the following intriguing theorem.
Theorem 1 (see [10]).Let the function This yields a n ≤ 4 − 2/n and na 2 n − a 2n−1 ≤ 2n − 1, for f ∈ Σ. Motivated by Theorem 1, bounds of a k ; k ≥ n and na 2  n − a 2n−1 can be investigated for various subclasses of Σ.In this paper, we investigate such bounds for the following general interesting subclass of Σ and for some of its special cases.Some improvements and generalizations for wellknown results will be obtained.
Σ,n η η ≥ 1 , if the following conditions are satisfied: where h, p ∈ P and The functions h ζ and p ζ can be specialized to provide interesting subclasses of analytic functions.If we set  3), is said to be in the class B Σ,n β, η , n ≥ 2 , if the following conditions are satisfied: where the function g is given by ( 7).
Similarly, it can be verified that the hypotheses of Definition 2 are satisfied for the choice where the functions ϕ and ψ are defined by This provides the following subclass of bi-univalent functions.
Definition 4. A function f ∈ Σ, in the form (3), is said to be in the class A Σ,n α, η , n ≥ 2 , if the following conditions are satisfied: where the function g is given by (7).
For the special case when n = 2, the class B h,p Σ,n η reduces to the class B h,p Σ η , which was introduced and studied by Xu et al. [37].However, for n = 2 and η = 1, the class B h,p Σ,2 1 was studied earlier by Xu et al. [38], and Definitions 3 and 4 have been defined by Frasin and Aouf [7], for the case when n = 2.
The following example shows that the subclasses A Σ,n α, η , B Σ,n β, η of the general class B h,p Σ,n η are not empty.

Example 5. Consider the function
Then, its inverse is given by With regard to specialization of the parameters of A Σ,n α, η and B Σ,n β, η which gives other examples of functions that show that those classes are not empty, see Adegani et al. [39].

Main Results
First, we find general coefficient bounds for functions in the class B h,p Σ,n η .
Setting n = 2 in Theorem 6 gives the following corollary.
Remark 8.The estimate (27) improves that given by Xu et al.
3 Journal of Function Spaces Proof.Let the functions h and p be defined as in (8).A computation shows, for k ≥ n, that By applying (31) and (32) to Theorem 6, we get the desired estimates.
The estimate of a n in Corollary 9 improves Corollary 3 in [32] and Theorem 1 in [33].Now, for the special case when n = 2, we have the following remark.
Note that Remark 10, for k = 3, reduces to Corollary 6 by Bulut [32].The estimates of a 2 and a 3 are much better than those given by Xu et al. ([37], Corollary 2) and Frasin and Aouf ([7], Theorem 3.2).Moreover, the estimate of a 2 which gives the range of β corresponds to the suitable bound of a 2 , which facilitates Corollary 11 in [40].Now, for the case whenever η = 1, Corollary 9 reduces to Theorem 3.2 in [10], for p = 1, as follows.
Remark 13.Note that the function given in Example 5, satisfies the conclusions of Remark 12. Indeed, in view of Remark 12, we find that

39
The following theorem introduces general coefficient bounds for functions in the class A Σ,n α, η .Theorem 14.Let f , in form (3), be in the class A Σ,n α, η , n ≥ 2 .Then, where ϕ and ψ are defined as in (11).It follows, for j ≥ n, that

Corollary 15. If
The estimates of a 2 and a 3 in Corollary 15 improve those given in Theorem 2.2 by Frasin and Aouf [7].In particular, for η = 1, the bounds improve the given estimates in Theorem 1 by Srivastava et al. [8].Also, the estimate of a 2 improves that given in Corollary 1 by Xu et al. [37] and Corollary 1 by Xu et al. [38].

Conclusion
Geometric function theory is a branch of complex analysis with a rich history that studies various analytical tools to study the geometric features of complex-valued functions.Due to the major importance of the study of the coefficients which plays an important role in the theory of univalent functions, the primary goal of this work is to determine coefficient bounds for certain general classes of bi-univalent 5 Journal of Function Spaces functions.Making use of Theorem 1 due to Al-Refai and Ali [10], a new method of estimating coefficients is applied, and interesting results that improve and generalize well-known estimates are obtained.The used technique may motivate other researchers to study other classes of bi-univalent functions and obtain new results.

8 where h k ≤ 2 −
2β and p k ≤ 2 − 2β, for every k ≥ n, and Re h ζ > β, Re p ζ > β, and ζ ∈ U , then the hypotheses of Definition 2 are satisfied, and we have the following subclass of bi-univalent functions.Definition 3. A function f ∈ Σ, in the form (