Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination

: In this paper, we introduce new subclasses R µ , α Σ , b , c ( λ , δ , τ , Φ ) and K µ , α Σ , b , c ( λ , δ , η , Φ ) of bi-univalent functions in the open unit disk U by using quasi-subordination conditions and determine estimates of the coefﬁcients | a 2 | and | a 3 | for functions of these subclasses. We discuss the improved results for the associated classes involving many of the new and well-known consequences. We notice that there is symmetry in the results obtained for the new subclasses R µ , α Σ , b , c ( λ , δ , τ , Φ ) and K µ , α Σ , b , c ( λ , δ , η , Φ ) , as there is a symmetry for the estimations of the coefﬁcients a 2 and a 3 for all the subclasses deﬁnd in our this paper.


Introduction
Let H be the class of analytic functions f defined in the open unit disk U = {z : |z| < 1} and normalized by conditions f (0) = 0, f (0) = 1. An analytic function f ∈ H has Taylor series expansion of the form: The well-known Koebe-One Quarter Theorem [1] states that the image of the open unit disk U under each univalent function in a disk with the radius 1 4 . Thus, every univalent function f has an inverse f −1 , such that Let Σ denote the class of all bi-univalent functions in U. Since f in Σ has the form (1), a computation shows that the inverse g = f −1 has the following expansion g(w) = f −1 (w) = w − a 2 w 2 + 2a 2 2 − a 3 w 3 − 5a 2 2 − 5a 2 a 3 + a 4 w 4 + . . . , w ∈ U.
Let B be the class of all analytic and invertible univalent functions in the open unit disk, but the inverse function may not be defined on the entire disk U, for f in H. An analytic function f is called bi-univalent in U if both f and f −1 are univalent in U.
In [9] Srivastava-Attiya introduced the following operator D µ,b : H −→ H, which has the following form b∈ C\{0, −1, −2, . . .}, µ∈ C, z ∈ U, f ∈ H. For f ∈ H, Carlson and Shaffer [10] defined the following integral operator T α f (z) by Define the convolution (or Hadamard product) of the operators D µ,b f (z) and T α f (z), which can be written as In the year 1970, the concept of quasi-subordination was first mentioned in [11]. For two analytic functions g and f in U, we say that the function f is quasi-subordinate to g in U, if there are analytic functions φ and F, with |φ(z)| ≤ 1, F(0) = 0 and |F(z)| < 1, such that f (z) = φ(z)g(F(z)), and denote this quasi-subordination by [12], as follows Note that if φ(z) = 1, then f (z) = g(F(z)), hence f (z) ≺ g(z) in U ( [13]). Furthermore, if F(z) = z, then f (z) = φ(z)g (z) and this case f is majorized by g, written as f (z) g(z) in U.
Ma and Minda [14], using the method of subordination of defined and studied classes S * (Φ) and G * (Φ) of starlike functions. See also [15,16] Now, consider an analytic and univalent function with a positive real part in U, symmetric with respect to the real axis and starlike with respect to Φ(0) = 1 and Φ (0) > 0. By S * Σ (Φ) and G * Σ (Φ) we denote the bi-starlike of Ma-Minda and bi-convex of the Ma-Minda type, respectively ( [17,18]).
In [17,19] Brannan and Taha get initial coefficient bounds for subclasses of bi-univalent functions. Later, Srivastava et al. [20] introduced and investigated subclasses of bi-univalent functions and get bounds for the initial coefficients. Also, Ali et al. [21] get the coefficient bounds for bi-univalent Ma-Minda starlike and convex functions. Some more important results on coefficient inequalities can be found in [12,[21][22][23].
Here, we discuss the improved results for the associated classes involving many of the new-known consequences.
We need the following Lemma to achieve the results.

The Subclass
and τ∈ C\{0}, if the following quasi-subordinations hold where g is the inverse function of f and z, w ∈ U.
For special values of parameters λ, δ, τ, µ, we obtain new and well-known classes.
where g is the inverse function of f and z, w ∈ U.
where g is the inverse function of f and z, w ∈ U.

Theorem 1. If f given by (1) belongs to the subclass
and where and where g is the inverse function of f and z, w ∈ U.
Taking δ = 0 in Theorem 1, we obtain the following corollary.

Corollary 1. Let f given by (1) belongs to the class
For λ = 1, we obtain
For special values of parameters η, δ, λ, µ, we obtain new and well-known classes.

Remark 3.
For η = 1 and δ∈ C\{0}, λ ≥ 0, a function f ∈ Σ defined in (1) is said to be in the class K µ,α Σ,b,c (λ, δ, η, Φ), if the following quasi-subordination conditions are satisfied: where g is the inverse function of f and z, w ∈ U. (1) is said to be in the class K Σ (λ, δ, η, Φ), if the following quasi-subordination conditions are satisfied: where g is the inverse function of f and z, w ∈ U.
Taking η = 1 in Theorem 2 we obtain the following corollary.

Discussion
We introduce new subclasses R µ,α Σ,b,c (λ, δ, τ, Φ) and K Σ (λ, δ, η, Φ) of bi-univalent functions in the open unit disk U by using quasi-subordination conditions and determine estimates of the coefficients |a 2 | and |a 3 | for functions of these subclasses. We obtained two new theorems with some new special cases for our new subclasses, and these results are different from the previous results for the other authors. Additionally, we discuss the improved results for the associated classes involving many of the new and well-known consequences. The results contained in the paper could inspire ideas for continuing the study, and we opened some windows for authors to generalize our new subclasses to obtain some new results in bi-univalent function theory.