Let be the class of analytic functions f(z) in the open unit disk

and represented by the normalized series:

We denote by the family of univalent functions in . (see, for details,[4, 27]). It is well known that every function has an inverse f⁻¹, defined by

and

The inverse function f⁻¹(w) is given by

The function is said to be bi-univalent in if (i) and (ii)f⁻¹(w) has an univalent analytic continuation to |w| < 1. Let ∑ be the class of bi-univalent functions in . Initial pioneering work on the class ∑ were done in [9, 16]. Srivastava et al. [26] mentioned some interesting examples of functions in the class ∑. Recently, Mishra and Soren [14] were add two more examples which are well demonstrated there in.

Špaček [19] and Libera [11] introduced the families of α-spirallike functions (-π2<α<π2) and α-spirallike functions of orderρ (-π2<α<π2,   0≤ρ≤1) respectively. Libera [11] completely settled the coefficient estimate problem for α-spirallike functions of order ρ. In this paper we introduce the families of α-bispirallike functions of order ρ and strongly α-bi-spirallike functions of order β. We find estimates for |a₂|, |a₃| and |a₄| for functions, of the form (1.1), in both these classes. Through out in this section also, we continue to denote by g the analytic continuation of the inverse of the function f to . We now have the following definitions:

Definition 1.1

The function f(z), given by (1.1), is said to be a member of α-SPΣβ, the class of strongly α-bi-spirallike functions of orderβ(∣α∣≤π2,0≤β<1), if each of the following conditions are satisfied:

and

Definition 1.2

The function f(z), given by (1.1), is said to be a member of , the class of α-bi-spirallike functions of orderρ(∣α∣≤π2,0≤ρ<1), if each of the following conditions are satisfied:

and

Furthermore, let ℘ be the class of analytic functions p(z) of the form:

and satisfy ℜ(p(z)) >0 ( ). We shall need this class to describe the classes α-SPΣβ and .

As follow up of the work of Mishra and Soren [14], at present there is renewed interest in the study of the class ∑ and its many new subclasses. For example see [1, 2, 3, 5, 7, 8, 10, 17, 18, 20, 21, 29, 30, 31]. Many researchers are still working upon finding an upper bound for a_n for the functions in subclasses of ∑. However, not much was known about the bound of the general coefficients a_n (n ≥ 4) of subclasses of bi-univalent functions up until the publication of the article by Mishra and Soren [14]. See [6, 13, 12, 15, 22, 23, 24, 25, 28]. For a brief history on the developments regarding the class ∑ see [26].

Motivated by the aforementioned work [14], in the present paper we have introduced two new subclasses of the function class ∑ and we find estimates for |a₂|, |a₃| and |a₄| for functions, of the form (1.1), when f∈α-SPΣβ and .

We state and prove the following:

Theorem 2.1

Let the function f(z), represented by the series (1.1), be in the class α-SPΣβ (∣α∣≤π2,0≤β<1). Then

and

Remark 2.2

Taking α = 0 in the above Theorem 2.1, we readily arrive at Mishra and Soren [14] of Theorem 2.1.

Theorem 2.3

Let f(z), given by (1.1), be in the class SPΣα(ρ)   (∣α∣≤π2,0≤ρ<1). Then

and

Remark 2.4

Taking α = 0 in the above Theorem 2.3, we readily arrive at Mishra and Soren [14] of Theorem 2.3.