Coefficient Estimates for Initial Taylor-Maclaurin Coefficients for a Subclass of Analytic and Bi-Univalent Functions Defined by Al-Oboudi Differential Operator

We introduce and investigate an interesting subclass 𝒩𝒫 Σ λ,δ(n, β; h) of analytic and bi-univalent functions in the open unit disk 𝕌. For functions belonging to the class 𝒩𝒫 Σ λ,δ(n, β; h), we obtain estimates on the first two Taylor-Maclaurin coefficients |a 2 | and |a 3 |.


Introduction
Let R = (−∞, ∞) be the set of real numbers, C the set of complex numbers, and We also denote by S the class of all functions in the normalized analytic function class A which are univalent in U.
For two functions and , analytic in U, we say that the function is subordinate to in U and write if there exists a Schwarz function , which is analytic in U with such that ( ) = ( ( )) , ( ∈ 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. In fact, the Koebe one-quarter theorem [3] ensures that the image of U under every univalent function ∈ S contains a disk of radius 1/4. Thus every function ∈ A has an inverse −1 , which is defined by In fact, the inverse function −1 is given by A function ∈ A is said to be bi-univalent in U if both and −1 are univalent in U. Let Σ denote the class of biunivalent functions in U given by (2). For a brief history and interesting examples of functions in the class Σ, see [4] (see also [5,6]). In fact, the aforecited work of Srivastava et al. [4] essentially revived the investigation of various subclasses of the bi-univalent function class Σ in recent years; it was followed by such works as those by Frasin and Aouf [7], Porwal and Darus [8], and others (see, e.g., [9][10][11][12][13][14][15][16][17]).
The Scientific World Journal 3 Remark 5. If we set in Definition 1, then the class NP , Σ ( , ; ℎ) reduces to the class denoted by NP Σ ( , ) which is the subclass of the functions ∈ Σ satisfying where ∈ (− /2, /2), ≥ 1, and the function is defined by (17).
Firstly, in order to derive our main results, we need the following lemma.
Lemma 7 (see [18]). Let the function ℎ( ) given by be convex in U. Suppose also that the function ( ) given by is holomorphic in U. If ( ) ≺ ℎ( ) ( ∈ U), then The object of the present paper is to find estimates on the Taylor-Maclaurin coefficients | 2 | and | 3 | for functions in this new subclass NP , Σ ( , ; ℎ) of the function class Σ.

A Set of General Coefficient Estimates
In this section, we state and prove our general results involving the bi-univalent function class NP , Σ ( , ; ℎ) given by Definition 1. with Proof. It follows from (16) that where ( ) ≺ ℎ( ) and ( ) ≺ ℎ( ) have the following Taylor-Maclaurin series expansions: respectively. Now, upon equating the coefficients in (36) and (37), we get From (40) and (42), we obtain The Scientific World Journal Also, from (41) and (43), we find that Since , ∈ ℎ(U), according to Lemma 7, we immediately have Applying (47) and Lemma 7 for the coefficients 1 , 2 , 1 , and 2 , from the equalities (45) and (46), we obtain respectively. So we get the desired estimate on the coefficient | 2 | as asserted in (34). Next, in order to find the bound on the coefficient | 3 |, we subtract (43) from (41). We thus get Upon substituting the value of 2 2 from (45) into (50), it follows that . (51) So we get On the other hand, upon substituting the value of 2 2 from (46) into (50), it follows that And we get Comparing the inequalities in (52) and (54) completes the proof of Theorem 8.

Corollaries and Consequences
By setting in Theorem 8, we have the following corollary.
By setting in Theorem 8, we have the following corollary. (59) By setting in Theorem 8, we have the following corollary.  (63) Remark 12. When = 0, Corollary 11 is an improvement of the following estimates obtained by Porwal and Darus [8].
Corollary 13 (see [8]). Let the function ( ) given by the Taylor-Maclaurin series expansion (2) be in the function class By setting in Theorem 8, we have the following corollary.
Remark 15. When = 0, Corollary 14 is an improvement of the following estimates obtained by Frasin and Aouf [7].
Corollary 19 (see [4]). Let the function ( ) given by the Taylor-Maclaurin series expansion (2) be in the function class