Coeﬃcient Estimates for a New Subclass of Analytic and Bi-Univalent Functions by Hadamard Product

: In this work, we introduce a new subclass of bi-univalent functions which is deﬁned by Hadamard product and subordination in the open unit disk and ﬁnd upper bounds for the second and third coeﬃcients for functions in this new subclass. Further, we generalize and improve some of the previously published results.


Introduction
Let A be a class of functions of the form f (z) = z + ∞ n=2 a n z n , (1.1) which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. Further, let S to denote the class of functions f ∈ A which are univalent in U.
Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk U. However, for each f ∈ S, the Koebe one-quarter theorem [11] ensures that the image of U under f contains a disk of radius 1 4 . Hence every function f ∈ S has an inverse f −1 , which is defined by and f (f −1 (w)) = w |w| < r 0 (f ); r 0 (f ) ≥ where g = f −1 and 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 + · · · . (1. 2) A function f ∈ A is said to be bi-univalent in U if both f and f −1 are univalent in U. Let Σ denote the class of bi-univalent functions in U given by (1.1).
In this work, we introduce a new subclass of bi-univalent functions which is defined by Hadamard product and find upper bounds for the second and third coefficients for functions in this new subclass. Besides, the estimates on the coefficients |a 2 | and |a 3 | presented in this work would generalize and improve some of results of Aouf et al. [4], Bulut [8], Ç aglar et al. [9], El-Ashwah [12], Frasin and Aouf [14], Murugusundaramoorthy [18], Orhan et al. [19], Porwal and Darus [20], Prema and Keerthi [21], Srivastava et al. [24], Srivastava et al. [25] and related works in this literature.

Preliminaries
In this section, we recall some definitions and lemmas that used in this work.
the Hadamard product (f * Θ)(z) of the functions f (z) and Θ(z) defined by a n c n z n . 11]). An analytic function f is said to be subordinate to another analytic function g, written as if there exists a Schwarz function w, which is analytic in U with such that f (z) = g(w(z)).

By (3.18), (3.19) and Lemma 2.3, we have
Now, we obtain the bound on |a 3 | according to µ from the above inequality Case 1. We suppose that let 0 ≤ µ < 1 thus we have Case 2. We let µ ≥ 1 thus we have which is the second part of assertion (3.21). So, from (3.21) and two above case, we obtain the desired estimate on |a 3 | given in (3.6). This completes the proof. ✷

Corollaries and Consequences
By setting m = 0 and Ω(z) = Θ(z) = z 1−z in Theorem 3.28, we obtain the following result which is an improvement of the estimates obtained for |a 2 | by Orhan et al. [19,Theorem 2.1].
Let the function f be given by (1.1) in the class NP λ,µ Σ (β; h). Then .
By setting m = β = 0 and Θ(z) = z 1−z in Corollary 4.5, we obtain the following result which is the estimates obtained for |a 3 | by Ç aglar et al. and By setting m = β = 0, µ = 1 in Corollary 4.5, we obtain the following result which is an improvement of the estimates obtained by El-Ashwah [12,Theorem 2].