Initial bounds for analytic and bi-univalent functions by means of (p, q)−Chebyshev polynomials defined by differential operator

In this paper, a subclass T ζ Σ (m, γ, λ, p, q) of analytic and bi-univalent functions by means of (p, q)−Chebyshev polynomials is introduced. Certain coefficient bounds for functions belong to this subclass are obtained. In addition, the Fekete-Szegö problem is solved in this subclass.


Introduction and preliminaries
Let A denote the class of functions of the form: f (z) = z + ∞ ∑ n=2 a n z n , (1.1) which are analytic in the open unit disk U = {z : |z| < 1}. Further, by S we shall denote the class of all functions in A which are univalent in U. It is well known that every function f ∈ S has an inverse f −1 , defined by and where f −1 (w) = w − a 2 w 2 + (2a 2 2 − a 3 )w 3 − (5a 3 2 − 5a 2 a 3 + a 4 )w 4 + · · · . A function f ∈ A is said to be in Σ the class of bi-univalent in U if both f (z) and f −1 (z) are univalent in U. Lewin [9] showed that |a 2 | < 1.51 for every function f ∈ Σ given by (1.1). Posteriorly, Brannan and Clunie [3] improved Lewin's result and conjectured that |a 2 | ≤ √ 2 for every function f ∈ Σ given by (1.1). The coefficient estimate problem for each of the following Taylor Maclaurin coefficients: It's still an open problem. Since then, there have been many researchers (see [2,5,6,7,11,12,14,15,13]) investigated several interesting subclasses of the class Σ and found non-sharp estimates on the first two Taylor-Maclaurin coefficients |a 2 | and |a 3 |. In fact, its worth to mention that by making use of the Faber polynomial coefficient expansions Jahangiri, Jay M., and Samaneh G. Hamidi [8] have obtained estimates for the general coefficients |a n | for bi-univalent functions subject to certain gap series.
Recently, Kızılateş, Naim and Bayram [16] 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 Mason and Handscomb was introduced [17], by τ q f (z) = f (qz),similarly, τ p,q f (z) = f (pqz).
First off, we present some special cases of the polynomials H p,q (z) : 1. For p = q = 1 and s = −1, we get the Chebyshev polynomials R n (x) of the second kind. 2. For p = q = s = 1 and x = x 2 , we get the Fibonacci polynomials F n (x). 3. For p = q = 1, s = 2y and x = 1 2 , we get the Jacobsthal polynomials J n+1 (y). 4. If p = q = s = 1, then we get the Pell polynomials P n+1 (x). Let w(z) and v(w) be two analytic functions in the unit disk U with w(0) = v(0) = 0, |w(z)| < 1, |v(z)| < 1, and suppose that Making use of the binomial series Frasin [4] defined the differential operator A ζ m,λ f (z) as follows: ζ a n z n ; ζ ∈ N, Using the relation (1.4), it is easily verified that By specializing the parameters we observe that, for m = 1, A ζ 1,λ defined by Al-Oboudi [1] and for m = γ = 1, A ζ 1,1 defined by Sȃlȃgean [10].

Coefficient bounds for the function class
We begin with the following result involving initial coefficient bounds for the function class T ζ Σ (m, γ, λ, p, q).
By taking λ = 1 in Theorem 3.1, we have