Univalent Functions by Means of Chebyshev Polynomials

*e primary motivation of the paper is to define a new class Chδ(α, β, c) which consists of univalent functions associated with Chebyshev polynomials. For this class, we determine the coefficient bound and convolution preserving property. Furthermore, by using subordination structure, two new subclasses of Chδ(α, β, c) are introduced and denoted by M(λ1, λ2, s) and N(λ1, λ2, s), respectively. For these subclasses, we obtain coefficient estimate, extreme points, integral representation, convexity, geometric interpretation, and inclusion results. Moreover, we prove that, under some restrictions on parameters, Chδ(α, β, c) � N(λ1, λ2, s).


Introduction
Let A be the class of analytic univalent functions in the open unit disk: with Taylor expansion series of the form Also, denote by S the class of univalent functions which are normalized by f(0) � f ′ (0) − 1 � 0, see [1,2]. Furthermore, suppose that N be the subclass of A consisting of functions with negative coefficients of the type: e significance of Chebyshev polynomials in numerical analysis is very important in both practical and theoretical points of view.
ere are four kinds of such polynomials. Many researchers consider orthogonal polynomials of Chebyshev and obtain many interest results.

Main Results
In this section, we introduce a sharp coefficient bound for V(z) ∈ Ch δ (α, β, c). Also, convolution preserving property under parameters α and β is proved.
Conversely, let V(z), defined by (15), be in the class Since, for any z, |Rez| < |z|, then By letting z ⟶ 1 through real values, we obtain and this completes the proof.
□ Remark 1. We note that the function, shows that inequality (17) is sharp.
Proof. (i) It is sufficient to show that By using the Cauchy-Schwarz inequality from (31), we obtain Hence, we find the largest α * such that ∞ k�2 k(k + 2cδ)sin 2 (k + 1)θ is inequality holds if or equivalently (ii) By using the same techniques as in the part (i), we can easily prove the part (ii), so the proof is complete.

Subclass of Ch δ (α, β, γ) and Their Geometric Properties
In this section, we introduce two new subclasses of Ch δ (α, β, c) and conclude their geometric properties.

Journal of Function Spaces
which implies that Now, we choose the values of z on the real axis, and letting z ⟶ 1 − , we get the required result.

(45)
By eorem 3, it is enough to show that (33) holds true, which is possible when or equivalently Since k starts from 2, then k − 1 ≥ 1, and hence, from the last inequality, we obtain the required result.

□
In the next theorems, we prove the inclusion property and convex combination concept. Also, extreme points and integral representation are introduced. Theorem 6. Let 0 ≤ s 2 < s 1 < 1; then, Proof. Suppose f ∈ N(λ 1 , λ 2 , s)[s 1 ]; then, by eorem 3, we have We have to prove However, the last inequality holds true if and this inequality by hypothesis (s 2 < s 1 ) definitely holds true, so the proof is complete. □ Theorem 7. N(λ 1 , λ 2 , s) is a convex set.