The Extension of Chebyshev Polynomial Bounds Involving Bazilevic Function

Objectives: To propose a new class of bi-univalent function based on Bazilevic Sakaguchi function using the trigonometric polynomials T n ( q ; e i q ) and to ﬁnd the Taylor – Maclaurin coeﬃcient inequalities and Fekete – Szego inequality for upper bounds. Methods: The Chebychev’s polynomial has vast applications in GFT. The powerful tool called convolution (Or Hadamard product), subordination techniques are used in designing the new class. In establishing the core results, derivative tests, triangle inequality and appropriate results that are existing are used. Findings: The trigonometric polynomials are applied and a class of Bi-univalent functions P a ; b ; c S ( l ; t ; q ; q ) involving Bazilevic Sakaguchi function is derived. More over, the maximum bounds for initial coeﬃcients and Fekete-Szego functional for the underlying class are computed. This ﬁnding opens the door to young researchers to move further with successive coeﬃcient estimates and related research. Novelty: In recent days, several studies on Chebyshev’s polynomial are revolving around univalent function classes among researchers. But in this article a signiﬁcant amount of interplay between Chebyshev’s polynomial and Bazilevic Sakaguchi function associated with Bi-univalent functions is clearly established.

A function f ∈ S is said to be Bazilevic function if it satisfies (see (7) ): This class of the function was denoted by B λ .Consequently whenλ = 0, the class of starlike function is obtained.In recently, P.Lokesh et al (8) investigated the inequalities of coefficient for certain classes of Sakaguchi type functions that satisfy geometrical condition as for complex numbers s, t but s ̸ = t and α (0 ≤ α < 1) .The convolution or Hadamard product of two functions f , g ∈ Ais defined by f * gand is defined by where f is given by (1) and g (z ) = e 2iθ + q 2 e −2iθ + q....
The obtained results for q = 1give the corresponding ones for Chebyshev polynomials of the second kind.The classical Chebyshev polynomials which are used in this paper, have been in the late eighteenth century, when was defined using de Moivre's formula by Chebyshev(refer (9) ).Such polynomials as (for example) the Fibonacci polynomials, the Lucas polynomials, the Pell polynomials and the families of orthogonal polynomials and other special polynomials as well as their generalizations are potentially important in the fields of probability, statistics, mechanics and number theory (10)(11)(12)(13)(14) 2 Methodology In the present work, the convolution operator I a,b,c due to Hohlov (refer (15,16) , which is special case of the Rajavadivelu Themangani et al (refer (17) ) is recalled.
https://www.indjst.org/where (a) n is the Pochhammer symbol (or the shifted factorial) given by Now, let us consider a linear operator introduced by Isra Al-shbeil et al (refer (18) ) and It is observed that, for a function f of the form (1), where In this paper, a new class of bi-univalent function based on Bazilevic Sakaguchi function using the trigonometric polynomials T n ( q, e iθ ) is established.Furthermore, the coefficient bounds and Fekete -Szego inequalities are also derived for this class.1) is said to be in the class P a,b,c Σ (λ , τ, q, θ )if it satisfies the following conditions, where the function g = f −1 .By taking the parameters λ = 0 and τ = 0, which was introduced by Sahsene Altinkaya et al (refer (19) ).
Taking the parameters λ = 0and τ = 0in the Theorem 1, we get the following remark.
, and which was investigated by Sahsene Altinkaya et al (19) .Fekete -Szego inequality for the function classP a,b,c Σ (λ , τ, q, θ ) In this section, we provide Fekete -Szego inequalities for function in the class P a,b,c Σ (λ , τ, q, θ ).This inequality is given in the following theorem.
Proof.From the equation ( 18) and the equation (19), we observe that Then, in view of equation ( 10), we get This evidently completes the proof of Theorem 2.

Conclusion
In the present investigation, a new class of bi univalent function based on Bazilevic Sakaguchi function using the trigonometric polynomials T n ( q, e iθ ) is obtained in the open unit disc.Furthermore, belonging to this class, the Taylor -Maclaurin coefficient inequalities and the well knownFekete -Szego inequalities are also derived.These findings can further be improved by finding sharpness.Moreover, Hankel Determinants and Toeplitz determinants for various integral orders can be computed in the future.
be the set of real numbers.C be the complex numbers andN := {1, 2, 3, ...} = N 0 / (0}be the set of positive integers.Let ∆ = (z ∈ C : (z| < 1} be open unit disc in C. A well known, the trigonometric polynomials T n ( q, e iθ ) are expressed by the generating function