Applications of ( p, q ) -Gegenbauer Polynomials on a Family of Bi-univalent Functions

In this work, we investigate the ( p, q )-Gegenbauer polynomials for a class of analytic and bi-univalent functions deﬁned in the open unit disk, with respect to subordination. We give an elementary proof to establish some estimates for the coeﬃcient bounds for functions in the new class. We conclude the study by giving a result of the Fekete-Szeg¨o theorem. A corollary was given to show some results of some subclasses of our new class.

The Koebe one-quarter theorem (see [30]) declares that the image domain of every function f ∈ S contains a disk of radius 1/4.This means that every function f ∈ S has an inverse function f −1 which can be defined by A function f ∈ A is said to be bi-univalent in ∆ if both f and f −1 are univalent in ∆.Let B denote the class of analytic and bi-univalent functions in ∆.In 1967, Lewin [14] introduced the class of bi-univalent functions and declared that every bi-univalent function has upper bound |a 2 | < 1.51.Some other results, examples, properties, definitions and some historical background, are archived in [10-13, 16, 21, 25, 28, 29].
Suppose c(z) is an analytic function, then is called the class of Schwarz functions.Let ≺ denote subordination, so if j, J ∈ A, j ≺ J if j(z) = J(c(z)) and z ∈ ∆.However, if J is a univalent function, then for z ∈ ∆, j(z) ≺ J(z) if, and only if, j(0) = J(0) and j(∆) ⊂ J(∆).
Lately, orthogonal polynomials have been a focal point of studies in the field of geometric function theory.Some works in this direction can be found in [3-6, 9, 10, 18, 20-22, 24, 28].The Chebyshev polynomials of the second kind is the natural generalization of Chebyshev polynomials of the first kind.It can be used in different areas of mathematics such as in theory of approximation, linear algebra, discrete analysis, representation theory and physics.For n ∈ {2, 3, 4, . ..}, 0 < q < p 1 and a variable s, the generating function of (p, q)-Chebyshev polynomials of the second kind is defined by where η q f (z) = f (qz) is the Fibonacci operator defined by Mason [15] and in a similar manner, Kizilatecs et al. [10] defined the operator η p,q f (z) = f (pqz).The recurrence relation for the (p, q)-Chebyshev polynomials of the second kind is defined by with initial values U 0 (x, s, p, q) = 1 and U 1 (x, s, p, q) = (p + q)x.Let α be a nonzero real constant, the generating function for the Gegenbauer polynomials is defined by where x ∈ [−1, 1] and z ∈ ∆.For a fixed x, the function G α is analytic in ∆ so it can be expanded in a Taylor's series as where V α n (x) is known as the Gegenbauer polynomials of degree n.Obviously, G α generates nothing when α = 0, therefore the generating function of the Gegenbauer polynomials is set to The Gegenbauer polynomials can as well be defined by the relation which produce some initial values expressed as Observe that from (1.10), if α = 1, then we get the second kind Chebyshev polynomials and if α = 1 2 , then we get the Legendre polynomials.See [2,19,23,31,32] for some details.
It is interesting to know that (1.9) can be generalized by the recurrence relation where 0 < q < p 1, s is an arbitrary variable and the initial values are given by We remark that (1.12) are the (p, q)-Gegenbauer polynomials from which for α = 1, we get the (p, q)-Chebyshev polynomials and for α = 1 2 , we get the (p, q)-Legendre polynomials.Further, a careful variation of the involving parameters show that we will get the listed polynomials in Remark 1.1.

Associated Lemmas
Let c(z) be as defined in (1.3), then the following lemmas hold to prove our results.
Equality holds for functions c(z) = z or c(z) = z 2 .
Remark 3.1.The following are subclasses of class BS(s, t, G).
2. If we set α = 1 = s and t = 0, then class BS(s, t, G) becomes class BS(C p,q ) which consists of bi-starlike functions that are subordinate to (p, q)-Chebyshev function and defined by the conditions zf (z) f (z) ≺ C p,q (x, z) and wF (w) F (w) ≺ C p,q (x, w) z, w ∈ ∆.
3. If we set α = p = q = s = 1 and t = 0, then class BS(s, t, G) becomes class BS(C) which consists of bi-starlike functions that are subordinate to Chebyshev function and defined by the conditions In this work, we use the (p, q)-Gegenbauer polynomials to define two new classes of analytic-bi-univalent functions that are associated with them.The initial coefficient estimates were afterward established for the two classes.

Main Results
In what follows, let all the parameters be as declared in Section 3 unless otherwise mentioned.Thus, the established results are as follows.
Theorem 4.1.Let f ∈ B be a member of BS(s, t, G).Then and for µ ∈ R we get Proof.Let f ∈ BS(s, t, G), then there exists the analytic functions and (s − t)wF (w) and (4.5) Also, using (1.1) and some simplifications in (4.2) we get and using ( and some simplifications in (4.3) we get In view of the corresponding equations in (4.6) and (4.7) we get Now if we add (4.8) and (4.10) we will get Also, if we add the squares of (4.8) and (4.10), we will get Likewise if we add (4.9) and (4.11), we will get and putting (4.14) into (4.15)simplifies to x, s, p, q)} (4.16) so that using (1.12), taking modulus of both sides and applying Lemma 2.1 give the required result.Now if we subtract (4.11) from (4.9), we get and using (1.12), we get so that taking modulus of both sides and applying Lemma 2.1 give the required result.
A special case of Theorem 4.1 is given in the following Corollary.