[Paper Review] Fekete-Szego Inequality For Analytic And Bi-univalent Functions Subordinate To (p; q)-Lucas Polynomials

Ala Amourah|arXiv (Cornell University)|Mar 30, 2020

Analytic and geometric function theory15 references4 citations

TL;DR

This paper introduces a new subclass of analytic and bi-univalent functions subordinate to (p,q)-Lucas polynomials and solves the Fekete-Szegő problem for this class. By employing coefficient bounds derived from the recurrence relations of (p,q)-Lucas polynomials and applying subordination principles, the authors establish sharp inequalities for the functional |a₃ − υa₂²|, generalizing known results for bi-starlike functions and extending coefficient estimates to a broader class of functions with parameterized differential subordination.

ABSTRACT

In the present paper, a subclass of analytic and bi-univalent functions by means of (p; q)- Lucas polynomials is introduced. Certain coefficients bounds for functions belonging to this subclass are obtained. Furthermore, the Fekete-Szego problem for this subclass is solved.

Motivation & Objective

• To define a new subclass of analytic and bi-univalent functions using (p,q)-Lucas polynomials.
• To derive sharp coefficient bounds for the Taylor-Maclaurin coefficients a₂ and a₃ in this new class.
• To solve the Fekete-Szegő problem for functions in this subclass, providing estimates for |a₃ − υa₂²|.
• To generalize known results for bi-starlike and bi-convex functions by incorporating (p,q)-Lucas polynomial subordination.

Proposed method

• The class 𝔅Σμ(α,λ,δ) is defined via a differential subordination condition involving (p,q)-Lucas polynomials Lp,q,k(x).
• Coefficient bounds for a₂ and a₃ are derived using the inverse function expansion and subordination techniques.
• The Fekete-Szegő inequality is established by analyzing the expression a₃ − υa₂² through transformation into variables r₂, s₂, r₁, s₁ representing coefficients of f and f⁻¹.
• The solution relies on the recurrence and generating function of (p,q)-Lucas polynomials, with Lp,q,1(x) = p(x) and Lp,q,2(x) = p²(x) + 2q(x).
• The bounds are derived by splitting the solution into cases based on the value of |φ(υ,x)| relative to 1/(2(μ + 2λ + 2ξδ)).
• Special cases are obtained by setting μ = δ = 0, λ = 1, recovering known results for bi-starlike functions.

Experimental results

Research questions

• RQ1 What are the sharp coefficient bounds for the second and third Taylor-Maclaurin coefficients of analytic and bi-univalent functions subordinate to (p,q)-Lucas polynomials?
• RQ2 How does the Fekete-Szegő functional |a₃ − υa₂²| behave for functions in this new class?
• RQ3 What is the impact of the parameters μ, λ, δ, and the (p,q)-Lucas polynomial structure on the coefficient estimates?
• RQ4 How do the derived inequalities generalize known results for bi-starlike and bi-convex functions?
• RQ5 What are the limiting cases of the Fekete-Szegő inequality when υ = 1 or when μ = δ = 0, λ = 1?

Key findings

• The sharp upper bound for |a₂| is given by 2|p(x)|√|p(x)| / √| (μ+2λ)[1+μ+12δ/(2λ+1)]p²(x) − 2(μ+λ+2ξδ)²(p²(x)+2q(x)) |.
• The bound for |a₃| is bounded by p²(x)/ (μ+λ+2ξδ)² + |p(x)| / (μ+2λ+2ξδ).
• For the Fekete-Szegő functional, |a₃ − υa₂²| ≤ |p(x)| / (μ+2λ+2ξδ) when |υ−1| ≤ |Υ(x)| / [2(μ+2λ+2ξδ)], with Υ(x) defined in the paper.
• When |υ−1| ≥ |Υ(x)| / [2(μ+2λ+2ξδ)], the bound becomes 2|p(x)|³|1−υ| / |p(x)Υ(x)|.
• In the special case μ = δ = 0, λ = 1, the bound reduces to |a₃ − υa₂²| ≤ |p(x)| if |υ−1| ≤ |q(x)/p(x)|, and 2|p(x)|³|1−υ| / |4q(x)| otherwise.
• For υ = 1, the bound simplifies to |a₃ − a₂²| ≤ |p(x)| / (μ+2λ+2ξδ), which generalizes the classical Fekete-Szegő inequality for bi-univalent functions.