The Extension of Chebyshev Polynomial Bounds Involving Bazilevic Function

Abstract

Objectives: To propose a new class of bi-univalent function based on Bazilevic Sakaguchi function using the trigonometric polynomials T n ( q, e iθ ) 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 Σ (λ , τ, q, θ ) 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.