ESTIMATION OF COEFFICIENT BOUNDS FOR THE SUBCLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH CHEBYSHEV POLYNOMIAL

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The most essential and the needed concept used by the theory of complex function is the Quasi subordination. The subordination along with the majorization concepts are getting collaborated with the help of this Quasi subordination concept. In this article, a novel subclass consisting of univalent analytic functions are investigated, analysed and reviewed. The Chebyshev polynomials that is associated with the open unit disk is defined. Further the estimates are found in these classes having the coefficients of functions by utilizing the benefits of Chebyshev polynomials. Then the Fekete-Szegö inequalities are obtained which provides the results representing the associated new classes thereby briefly making the quasi-subordination to involve along with majorization results.


INTRODUCTION AND PRELIMINARY RESULTS
Let us denote A as the class of functions which is in the form (1) f (z) = z + ∞ ∑ n=2 a n z n , that seems to be analytic in the open unit disk U = {z ∈ C : |z| < 1} and is again normalized by the conditions f (0) = 0 = f (0) − 1. Also let us consider S as the subclass of A that consists of univalent functions in U.
Let as assume ω(z) as the class of analytic functions in the form ω(z) = ω 1 z + ω 2 z 2 + ω 3 z 3 + · · · , which satisfies with a suitable condition |ω(z)| < 1 in U, that is mentioned to be the class which is comprised of the Schwarz functions. By Recalling the subordination principle between the analytic functions, f (z) and g(z) are analytic in U. This could be mentioned that the function f (z) is considered as the subordinate to g(z), Schwarz function ω(z) gets existed, then f (z) = g(ω(z)), (z ∈ U). This make the functions are subordinate. This subordination condition is denoted by f ≺ g (or) f (z) ≺ g(z), (z ∈ U). Particularly the function g(z) is univalent in U, then the above mentioned subordination is much equivalent to the conditions f (0) = g(0), According to , the following subordination principle is stated as: Let us consider S * (φ ) as the deliberated class representing the starlike function f ∈ S for and C (φ ) be assumed as the class representing the convex function [21] analysed and reviewed these classes. Quasi-subordination is the extension of subordination which is introduced and studied by Robertson in [29,30]. The function f (z) is Quasi-subordinate onto the function g(z) in U whenever if arises the Schwarz function ω(z) along with an functions which is analytic given as ϕ(z) by satisfying the condition |ϕ(z)| < 1 as f (z) = ϕ(z)g(ω(z)) in U, which is also represented by By setting the value ϕ(z) ≡ 1 then the subordination reduction is typically done by quasisubordination. And by setting ω(z) = z, then we get f (z) = ϕ(z)g(z) and also consider that f (z) which is majorized by g(z) and is mentioned as f (z) g(z) in U. Hence the quasisubordination might be considered as the generalization by representing the notion of the subordination in addition with the majorization which emphasize the reputation of majorization.
The major role considered in the numerical analysis, applied mathematics and the approx- The Chebyshev polynomials are represented by T n (x) and U n (x) for all x ∈ [−1, 1] respectively are well defined by, and second kind, where n denotes the polynomial degree and x = cos θ .
Φ(z,t) can be written as are the Chebyshev polynomials of second kind.
Furthermore, we know that and The generating function of the first kind of Chebyshev polynomial T n (t), t ∈ [−1, 1], is given The first kind of Chebyshev polynomial T n (t) in addition with the second kind of Chebyshev polynomial U n (t) might gets related by considering the following relationship: Definition 1.1. A function f ∈ S is said to be in the class H q (α, β , Φ), α, β ∈ R if it satisfies the following condition Our current research proves that the usage of the Chebyshev polynomials might gets expanded to provide the basic coefficients of analytic functions in H q (α, β , Φ(z,t)), S q (α, Φ(z,t)), L q (α, Φ(z,t)). We also have to find the Fekete-Szegö estimation for the above defined class associated with quasi-subordination and majorization.
The lemma that is following in regards to the coefficients of functions in ω(z) might needed for proving our main results.  Then ω 2 − µω 2 1 ≤ max{1, |µ|}, for any complex number µ. The result is sharp for the function ω(z) = z or ω(z) = z 2 .
By Applying the same technique as in Theorem 2.1 for the classes S q (α, Φ) and L q (α, Φ), the following Theorems are obtained.
Theorem 2.2. Let ϕ(z) be given in (2) and if f (z) given by (1) belongs to S q (α, Φ), then and for some µ ∈ C Theorem 2.3. Let ϕ(z) be given in (2) and if f (z) given by (1) belongs to L q (α, Φ), then and for some µ ∈ C Equating the coefficient of z and z 2 we have the following coefficients (10) Since µ is a complex number, from (10) and (11), we get Substituting the value of d 1 from (7) in (12), which implies that If d 0 = 0, the equation (12) becomes .
By Implementing the same method as in Theorem 3.1 for the classes S q (α, Φ(z,t)) and L q (α, Φ(z,t)), we get the following Theorems.
Theorem 3.2. Let ϕ(z) be given in (2) and if f (z) given by (1) satisfies the condition and for some µ ∈ C Theorem 3.3. Let ϕ(z) be given in (2) and if f (z) given by (1) satisfies the condition

CONCLUDING REMARK
From the above researches lot of various mind blowing consequences that generates the results that gets asserted in the theorems proven above might gets derived by particularly specialising the parameters that provides the results.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.