Faber Polynomial Coefficient Estimates for Subclass of Analytic Bi-Bazilevic Functions Defined by Differential Operator

In this work,  an explicit formula for a class of Bi-Bazilevic univalent functions involving differential operator is given, as well as the determination of upper bounds for the general Taylor-Maclaurin coefficient of a functions belong to this class, are established Faber polynomials are used as a coordinated system to study the geometry of the manifold of coefficients for these functions. Also determining bounds for the first two coefficients of such functions. 
         In certain cases, our initial estimates improve some of the coefficient bounds and link them to earlier thoughtful results that are published earlier. 
 


Introduction:
Let  be the class of all functions () as the following form , ( ∈ ) which are analytic and normalized in the open unit disk  = { ∈ ℂ ∶ || < 1}.Also, let  be the subclass of  consist of all functions that are univalent functions in .A function  ∈  has an inverse  −1 is defined as follows In fact, if  =  −1 is the inverse of the function  ∈ , then  has a Maclaurin series expansion in some disk about the origin which is given by . (1.3) Let the class of analytic functions of   be ()  =   + ∑   () +−1 .
Remark 1.1 It is easily to see from (1.7), that the operator reduces to several known differential operators by giving specific values to the parameters which have been studied by following earlier authors for instance: I. For  = 1,  = 0 and  = 1, we get to the operator which was introduced by Al-Obouudi (1).II.For  = 1,  = 0  = 1, and  = 1, we get to the operator which was introduced by Salagean (2). and The above conditions are equivalent to respectively for some ( > 0, , ,  ≥ 0 ;  ∈ ℕ 0 = ℕ ∪ {0};  > 0,  ∈ ) and 0 ≤  < 1,  =  −1 is defined by (1.3).Estimate on the coefficients bounds of classes of meromorphic and univalent functions were widely researched in the literature.For instance, in 1948 Schiffer (5) proved that the estimate | 2 | ≤ 2 3 for meromorphic and univalent functions  with  0 = 0 and Duren (6) Hamidi S. G., Janani T, Murugusundaramoorthy G., and Jahangiri J. M. ( 7 The real difficulty emerges when the Biunivalency conditions are forced on the meromorphic functions  and its inverse  =  −1 .The sudden and bizarre conduct of the coefficients of meromorphic functions  and their inverse  =  −1 , prove the investigation of the coefficient bounds for Bi-univalent functions to be extremely challenging. In order to extend the results of Hamidi S. G., Janani T, Murugusundaramoorthy G., and Jahangiri J. M. ( 7) to a general class of meromorphic Biunivalent functions, we use the instrument of the well-known Faber polynomial expansions to determine estimates for a general subclass of analytic Bi-Bazilevic functions.Furthermore, we prove the unperdictability of the early first two cofficients of such Bi-Bazilevic functions that belong to this class which is the best estimate in the literature.

It is clear that
Evidently,    ( 1 ,  2 , … ,   ) =  1  ; this means that the first polynomials and last polynomials are    =  1  ,   1 =   .For instance the first three elements of  −1 − are: A similar Faber polynomial expansion formula holds for the coefficients of , the inverse function of .The Faber polynomials presented which play an important role in different areas of mathematical sciences, particularly in geometric function theory (Schiffer (5)).The recent interest for the calculus of Faber polynomials, particularly when it includes  =  −1 , the inverse of  (see (10), ( 11), ( 12), ( 13), (14), and ( 15)), flawlessly fits our case for the Bi-univalent functions.As a result, we can state the following.
Therefore, the left hand sides of the equations (2.3) and (2.4) can be expressed by and the same way, we obtain where  −1  is defined by (2.2) and Comparing the corresponding coefficient of (2.3) and (2.5), we get and similarly, from (2.4) and (2.6), we get Which under the assumption   = 0 for 2 ≤  ≤  − 1, we conclude that and (2.10) Note that, in virtue of Caratheodory Lemma (e.g. ( 8)), we have By taking the absolute value of equalities (2.9) and (2.10), we obtain the required bound The proof is complete.
The above Corollary 2.3 reduces to the result (( 16), Theorem 2.1) for analytic Bi-Bazilevic functions of order  and type ; 0 ≤  < 1which is studied by Jay M. Jahangiri and Samaneh G. Hamidi.
Note that Corollary 2.4 reduces to the result ((17), Theorem 1) for analytic Bi-univalent functions of order  and type  ; 0 ≤  < 1 which is studied by Jay M. Jahangiri and Samaneh G. Hamidi.
(2.18) Dividing by 2(1 − ) 2 and by applying the Caratheodory lemma.Consequently, we have This evidently completes the proof for theorem.

Definition 1 . 2
Now, using the differential operator Γ β,μ m,λ ()  to define a new class of Bi-Bazilevic type functions as follows: The function ()  which has the form (1.5) belongs to the class ℱ β,μ m,λ (, , ) satisfies the following conditions