On the Chebyshev Polynomial for a Certain Class of Analytic Univalent Functions

In this work, by considering the Chebyshev polynomial of the first and second kind, a new subclass of univalent functions is defined. We obtain the coefficient estimate, extreme points, and convolution preserving property. Also, we discuss the radii of starlikeness, convexity, and close-to-convexity.


Introduction
Let Δ be the open unit disk fz ∈ ℂ : jzj < 1g and A be the class of analytic functions in Δ, satisfying the normalized conditions: Thus, each f ∈ A has the following Taylor expansion: Furthermore, by S, we shall denote the family of all functions in A that are univalent in Δ. Denote by N the subclass of A consisting of functions with negative coefficients of the type: see [1].
Many researchers deal with orthogonal polynomials of Chebyshev, see [2,3] and [4]. The Chebyshev polynomials of first kind and the second kind are defined by respectively, where −1 < t < 1, t = cos θ, and k is the degree of polynomial.
The polynomial in (1) is connected by the following relations: We note that if t = cos θ, ð−π/3 < θ < π/3Þ, then Also, we have where are the Chebyshev polynomials of the second kind, see [5,6] and [7]. The generating function of the first kind of Chebyshev polynomial T k ðtÞ, t ∈ ½−1, 1 is given by For more details, see [8,9] and [10].
For two functions f and g, analytic in Δ, we say that f is subordinate to g in Δ, written if there exists a Schwarz function w which is analytic in Δ with such that f ðzÞ = gðwðzÞÞ, (z ∈ Δ), see [11]. Also, if g is univalent in Δ, then Furthermore, if f ðzÞ = z − ∑ ∞ k=2 a k z k and gðzÞ = z − ∑ ∞ k=2 b k z k , then the Hadamard product (or covolution) of f and g is defined by Now, we consider the following functions which are connected with the Chebyshev polynomial of the first and second kind: where f ðzÞ = z − ∑ ∞ k=2 a k z k ∈ N and " * " denotes the Hadamard product.
With a simple calculation we conclude that QðzÞ belongs to N and it is of the form: where −π/3 < θ < π/3 and t = cos θ.
Equation (19) is equivalent to the following inequality:

Main Results
In this section, we introduce a sharp coefficient bound for the class E λ γ ðα, βÞ. Also, the convolution preserving property is investigated.
Proof. Let the inequality (21) holds and z ∈ ∂Δ = fz ∈ ℂ : jzj = 1g. We have to prove that (19) or equivalently (20) holds true. But we have 2 Journal of Function Spaces By putting z ∈ ∂Δ and the above expression reduces to Since H − α = ðβ − αÞð1 − γÞ, by using inequality (21), we get Y ≤ 0, so Q ∈ E λ γ ðα, βÞ. To prove the converse, let Q ∈ E λ γ ðα, βÞ, thus for all z ∈ Δ. By Re ðzÞ ≤ jzj for all z ∈ Δ, we have By letting z ⟶ 1, through positive values and choose the values of z such that zQ ′ ðzÞ/f λ ðzÞ is real, we have and this completes the proof.
Remark. We note that the function: shows that the inequality (21) is sharp.

Theorem 2. Let
be in the class E λ γ ðα, βÞ, then ðQ 1 * Q 2 ÞðzÞ belongs to E λ γ ðα,βÞ, wherẽ Proof. It is sufficient to show that By using the Cauchy-Schwarz inequality, from (21), we obtain 3 Journal of Function Spaces Here, we find the largest k such that or equivalently for k ≥ 2,