Coefficient Bounds for a New Subclasses of Bi-Univalent Functions Associated with Horadam Polynomials

: In this work we present and investigate three new subclasses of the function class of bi-univalent functions in the open unit disk defined by means of the Horadam polynomials. Furthermore, for functions in each of the subclasses introduced here, we obtain upper bounds for the initial coefficients | | and | | . Also, we debate Fekete-Szegӧ inequality for functions belongs to these subclasses.


Introduction
Symbolized by the function class of the shape: The Koebe One-Quarter Theorem [ 4 ] shows that the image of includes a disk of radius ¼ under each function from . Thereby each univalent function of this kind has an inverse which fulfills The function is considered bi-univalent in if together and are univalent in Indicated by the Taylor-Maclaurin series expansion (1), the class of all bi-univalent functions in can be symbolized by . In the year 2010, Srivastava et al. [ 10 ] refreshed the study of various classes of bi-univalent functions. Moreover, many penmans explored bounds for different subclasses of bi-univalent functions ( see, for example [ 3,5,6,11 ] In special, if the function is univalent in , the above subordination is equivalent to The following recurrence relation gives the Horadam polynomials ( ) ( see ( 8 ) ) are some real constants. The characteristic equation of repetition relationship (3) is . There are two real roots of this equation

√ √
The generating function of the Horadam polynomials ( ) is indicated by It should be noted that for specific values of , the Horadam polynomial ( ) leads to different polynomials, among those, we list a few cases here ( see, [ 7 , 8 ], for more details ) : where the function is indicated by (2) and is real constant.

Remark 1
For , the class ( ) shortens to the class presented and investigated by Alamoush [ 2 ].
For , the class ( ) shortens to the class ( ) presented and investigated by Abirami et al. [ 1 ].
Theorem 1 Let the function indicated by (1) be in the class ( ). Then and for some , and . Then there are two holomorphic function indicated by and Or, in equivalent way, (10) and (11), we attain It follows from (12) and (13) that (14) and (16), we find that If we add (15) to (17), we get Next, if we deduct (17) from (15), we get In view of (18) and (19), equation (22) becomes Now, with the help of equation (3), we deduce that Finally, by using (21) and (22) for some , we get Thus, we conclude that and with respect to (3), it evidently completes the proof of the theorem (1).

Remark 2
If we put in Theorem (1), we get the outcomes which were indicated by Alamoush [ 2 ]. In addition, if we put in Theorem (1), we get the outcomes which were indicated by Abirami et al. [ 1 ].

Coefficient bounds and Fekete-Szegӧ inequality for the class ( )
Definition 2 A function is said to be in the class ( )for and , if the following conditions of subordination are satisfied: where the function is indicated by (2) and is real constant.

Remark 3
For , the class ( ) shortens to the class ( ) introduced and investigated by Srivastava et al. [ 9 ]. and Or, in equivalent way, From the equations (28) and (29), we attain It follows from (30) and (31) that (32) (2), we get the outcomes which were indicated by Srivastava et al. [ 9 ].

Coefficient bounds and Fekete-Szegӧ inequality for the class ( )
Definition 3 A function is said to be in the class ( ) for * + and , if the following conditions of subordination are satisfied: where the function is indicated by (2) and is real constant.  (48) and (49) that (50) and (52), we find that