COEFFICIENT ESTIMATES FOR BI-UNIVALENT FUNCTIONS IN CONNECTION WITH (p,q) CHEBYSHEV POLYNOMIAL

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this present work, authors are introduced a new subclass of bivalent functions SΣ(α,x, p,q) with respect to symmetric conjugate points in the open unit disc U related to (p,q) polynomials. Further the initial bounds of the subclass and the well known Fekete-Szegö inequality are determined.


INTRODUCTION
Let R=(−∞, ∞)be the set of real numbers, C be the set of complex numbers and in the open disc U = {z : z ∈ C : |z| < 1}. Further, let S denote the class of functions in A which are also univalent in U.
The well-known Koebe one-quarter theorem [2] ensures that the image of U under every univalent function f ∈ A contains a disc of radius 1/4. Hence every univalent function f has an inverse f −1 satisfying f −1 ( f (z)) = z, (z ∈ U) and Σ denote the class of bi-univalent functions in U given by (1.1). For example, functions in the class Σ are given below [8]: In 1967, Lewin [5] introduced the class Σ of bi-univalent functions and shown that |a 2 | < 1.51. In 1969, Netanyahu [7] showed that max f ∈Σ |a 2 | = 4/3 and Suffridge [9] have given an example of f ∈ Σ for which |a 2 | = 4/3. Later, in 1980, Brannan and Clunie [1] improved the result as |a 2 | ≤ √ 2. In 1985, Kedzier-awski [3] proved this conjecture for a special case when the function f and f −1 are starlike. In 1984, Tan [10] proved that |a 2 | ≤ 1.485 which is the best estimate for the function in the class of bi-univalent functions.
For any integer n ≥ 2 and 0 < q < p ≤ 1, the (p,q)-Chebyshev polynomials of the second kind is defined by the following recurrence relations: U n (x, s, p, q) = (p n + q n )xU n−1 (x, s, p, q) + (pq) n−1 sU n−2 (x, s, p, q) with the initial values U 0 (x, s, p, q) = 1, U 1 (x, s, p, q) = (p + q)x and 's' is a variable. By Assuming various values of x,s,p and q we get some interesting polynomials as follows: • When x = x 2 , s = s, p = p and q = q, the (p, q)-Chebyshev polynomials of the second kind becomes (p, q)-Fibonacci polynomials.
• When x = x, s = -1, p = 1 and q = 1, the (p, q)-Chebyshev polynomials of the second kind becomes Second kind of Chebyshev polynomials.
Recently Kızılate¸s et al. [4] defined (p, q)-Chebyshev polynomials of the first and second kinds and derived explicit formulas, generating functions and some interesting properties of these polynomials.
The generating function of the (p, q)-Chebyshev polynomials of the second kind is as follows: where the Fibonacci operator τ q was introduced by Mason [6], τ q f (z) = f (qz). Similarly, Definition 1. For 0 < α ≤ 1, a function s ∈ σ is belong to the class S Σ (α, x, p, q) if it satisfies the following conditions where r = s −1 .

ESTIMATION OF INITIAL COEFFICIENTS & FEKETE-SZEGÖ INEQUALITY
Theorem 1. A function f ∈ Σ of the form (1.1) is said to be in the class S Σ (α, x, p, q), then Proof. Suppose that f ∈ S Σ (α, x, p, q), then from (1.3) and (1.4) and for its inverse map g = f −1 , we have = G p,q (ϕ(w)).
By using (2.12) in (2.13) we get, From (2.13) we acquired the result which is desired in (2.1).
Theorem 2. A function f ∈ Σ of the form (1.1) is said to be in the class S Σ (α, x, p, q), then , .

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.