Coefficient Bounds for Subclasses of Biunivalent Functions Associated with the Chebyshev Polynomials

We introduce and investigate new subclasses of biunivalent functions defined in the open unit disk, involving Sălăgean operator associated with Chebyshev polynomials. Furthermore, we find estimates of the first two coefficients of functions in these classes, making use of the Chebyshev polynomials. Also, we give Fekete-Szegö inequalities for these function classes. Several consequences of the results are also pointed out.


Introduction
Let A denote the class of analytic functions of the form normalized by the conditions (0) = 0 =   (0) − 1 defined in the open unit disk Let S be the subclass of A consisting of functions of form (1) which are also univalent in .Let S * () and K() denote the well-known subclasses of S, consisting of starlike and convex functions of order  (0 ≤  < 1), respectively.The Koebe one-quarter theorem [1] ensures that the image of  under every univalent function  ∈ A contains a disk of radius 1/4.Thus every univalent function  has an inverse  −1 satisfying  −1 ( ()) = , ( ∈ ) ,  ( −1 ()) =  (|| <  0 () ,  0 () ≥ 1 4 ) . ( A function  ∈ A is said to be biunivalent in  if both  and  −1 are univalent in .Let Σ denote the class of biunivalent functions defined in the unit disk .Since  ∈ Σ has the Maclaurin series given by (1), a computation shows that its inverse  =  −1 has the expansion An analytic function  is subordinate to an analytic function , written as () ≺ (), provided there is an analytic function  defined on  with (0) = 0 and |()| < 1 satisfying () = (()).
Chebyshev polynomials, which are used by us in this paper, play a considerable role in numerical analysis.We know that the Chebyshev polynomials are four kinds.The most of books and research articles related to specific orthogonal polynomials of Chebyshev family contain essentially results of Chebyshev polynomials of first and second kinds   () and   () and their numerous uses in different applications; see Doha [2] and Mason [3].
The well-known kinds of the Chebyshev polynomials are the first and second kinds.In the case of real variable  on (−1, 1), the first and second kinds are defined by where the subscript  denotes the polynomial degree and  = cos .We consider the function We note that if  = cos ,  ∈ (−/3, /3), then for all  ∈ Thus, we write where  −1 = sin( arccos )/ √ 1 −  2 , for  ∈ N, are the second kind of the Chebyshev polynomials.Also, it is known that The Chebyshev polynomials   (),  ∈ [−1, 1], of the first kind have the generating function of the form All the same, the Chebyshev polynomials of the first kind   () and the second kind   () are well connected by the following relationship: Several authors have introduced and investigated subclasses of biunivalent functions and obtained bounds for the initial coefficients (see [4][5][6][7][8][9][10]).In [11], making use of the Sȃlȃgean [12] differential operator, defined by and further for functions  of the form (4) Vijaya et al. [11] (also see [13]) defined and introduced two new subclasses of biunivalent functions.
In this paper, we use Chebyshev polynomials to obtain the estimates on the coefficients | 2 | and | 3 |.

Biunivalent Function Classes
Motivated by recent works of Altinkaya and Yalcin [14] (also see [15]) and recent studies on biunivalent functions involving Sȃlȃgean operator [11,13], in this section, we introduce two new subclasses of Σ associated with Chebyshev polynomials and obtain the initial Taylor coefficients | 2 | and | 3 | for the function classes by subordination.
We note that by specializing the parameters  and suitably fixing the values for  in Definition 1, we introduce (had not been studied so far) the following new subclasses of Σ as listed below.
In Definition 3, by specializing the parameters  and suitably fixing the values for  (had not been studied so far) the following new subclasses of Σ are as listed below.
be in the class S  Remark 10.Let  given by (1) be in the class K  Σ (Φ(, )).