Maclaurin Coefficient Estimates for a New Subclasses of (cid:2) -Fold Symmetric Bi-Univalent Functions

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Introduction
Denote by the family of functions that are holomorphicin the open unit disk = ∈ ℂ ∶ | | < 1" and normalized by the conditions 0 = $ 0 − 1 = 0 and having the form: 1.1 We also denote by + the subfamily of consisting of functions satisfying (1.1) which are also univalent in . According to the Koebe one-quarter theorem (see [8]), every function ∈ + has an inverse , which satisfies A function ∈ is said to be bi-univalent in if both and , are univalent in . We denote by < the family of bi-univalent functions in given by (1.1). For a brief history and interesting examples in the family < see the pioneering work on this subject by Srivastava et al. [22], which actually revived the study of bi-univalent functions in recent years. In a considerably large number of sequels to the aforementioned work of Srivastava et al. [22], several different subfamilies of the bi-univalent function family < were introduced and studied analogously by the many authors (see, for example, [1,2,5,10,11,12,16,18,19,20,25,26,27]).
For each function ∈ +, the function ℎ = = , ∈ , ∈ ℕ is univalent and maps the unit disk into a region with -fold symmetry. A function is said to befold symmetric (see [13]) if it has the following normalized form: We denote by + the family of -fold symmetric univalent functions in , which are normalized by the series expansion (1.3). In fact, the functions in the family + are onefold symmetric.
In [23] Srivastava et al. defined -fold symmetric bi-univalent functions analogues to the concept of -fold symmetric univalent functions. They gave some important results, such as each function ∈ < generates an -fold symmetric bi-univalent function for each ∈ ℕ. Furthermore, for the normalized form of given by (1.3), they obtained the series expansion for , as follows: where , = 6. We denote by < the family of -fold symmetric bi-univalent functions in . It is easily seen that for = 1, the formula (1.4) coincides with the formula (1.2) of the family <. Some examples of -fold symmetric bi-univalent functions are given as follows: with the corresponding inverse functions Recently, many authors investigated bounds for various subfamilies of -fold biunivalent functions (see [3,4,7,9,14,17,21,23,24,28,29]).
In order to prove our main results, we require the following lemma.

Coefficient Bounds for the Function Family
2. For = 0, the family Υ * , ; reduce to the family l , which was introduced by Liu and Wang [15].

Conclusion
This paper has introduced a new subfamilies Υ , ; and Υ * , ; of < and find estimates on the coefficients | | and | | for functions in each of these new subfamilies.