Fekete-Szegö Problem for a Subclass of Analytic Functions Associated with Chebyshev Polynomials

Chebyshev polynomials play a considerable role in numerical analysis ( [4], [8]). There are four kinds of Chebyshev polynomials. The first and second kinds of Chebyshev polynomials are defined by Tn(t) = cosnφ and Un(t) = sin(n+1)φ sinφ (−1 < t < 1) where n denotes the polynomial degree and t = cosφ. For a brief history of Chebyshev polynomials of the first kind Tn(t), the second kind Un(t) and their applications one can refer [1][16]. Now, we define a subclass of analytic functions in D with the following subordination condition:


Introduction
Let A be the class of the functions of the form: Let f and g be analytic functions in D. We define that the function f is subordinate to g in D and denoted by f (z) ≺ g(z) (z ∈ D) , if there exists a Schwarz function ω, which is analytic in D with ω(0) = 0 and |ω(z)| < 1 (z ∈ D) such that f (z) = g (ω(z)) (z ∈ D) .
So, according to [15], we write the Chebyshev polynomials of the second kind as following: The Chebyshev polynomials T n (t), t ∈ [−1, 1] of the first kind have the generating function of the form . There is the following connection by the Chebyshev polynomials of the first kind T n (t) and the second kind U n (t) : In 1933, Fekete and Szegö [6] obtained a sharp bound of the functional |a 3 − µa 2 2 |, with real µ (0 ≤ µ ≤ 1) for a univalent function f . Since then, the problem of finding the sharp bounds for this functional of any compact family of functions or f ∈ A with any complex µ is known as the classical Fekete-Szegö problem or inequality.
In this paper, we obtain initial coefficients |a 2 | and |a 3 | for subclass N (λ, β, t) by means of Chebyshev polynomials expansions of analytic functions in D. Also, we solve Fekete-Szegö problem for functions in this subclass.

Coefficient bounds for the function class N (λ, β, t)
We begin with the following result involving initial coefficient bounds for the function class N (λ, β, t) . and .

Corollary 2.2. Let the function f (z) given by (1.1) be in the class
.
If we choose β = 0 in Theorem 2.1, we get the following corollary.
For β = λ in Theorem 2.1, we obtain the following corollary.

be in the class
and .

5
For β = λ in Theorem 3.1, we obtain the following corollary.