[Paper Review] Second Hankel Determinant for certain class of bi-univalent functions defined by Chebyshev polynomials

Halit Orhan, N. Magesh+1 more|arXiv (Cornell University)|May 9, 2017

Analytic and geometric function theory29 references3 citations

TL;DR

This paper establishes sharp upper bounds for the second Hankel determinant $ |a_2a_4 - a_3^2| $ for a new subclass $ \mathcal{N}_{\sigma}^{\mu}(\lambda,t) $ of bi-univalent functions associated with Chebyshev polynomials in the open unit disk. By applying subordination principles and optimizing a real-valued function derived from coefficient constraints, the authors derive explicit bounds depending on parameters $ \lambda $, $ \mu $, and $ t $, improving upon existing results in the literature.

ABSTRACT

Making use of Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant of a subclass $\mathcal{N}_{σ}^μ\left( λ,t ight) $ of bi-univalent function class $σ.$

Motivation & Objective

• To investigate the second Hankel determinant $ |a_2a_4 - a_3^2| $ for a new class of bi-univalent functions defined using Chebyshev polynomials.
• To extend previous coefficient bounds for bi-univalent functions by incorporating parameters $ \lambda $, $ \mu $, and $ t $ associated with Chebyshev polynomial expansions.
• To provide explicit, sharp upper estimates for the second Hankel determinant under various parameter regimes.
• To improve upon existing results in the literature, particularly those of Mustafa [28], by refining the bounds through critical point analysis.

Proposed method

• The authors define a new subclass $ \mathcal{N}_{\sigma}^{\mu}(\lambda,t) $ of bi-univalent functions using subordination conditions involving Chebyshev polynomials of the first kind.
• They derive coefficient inequalities by applying the subordination principle to the function and its inverse, leveraging the series expansions of $ f(z) $ and $ f^{-1}(w) $.
• A real-valued function $ K(c,t) $ is constructed based on the coefficients $ a_2, a_3, a_4 $, and the parameters $ \lambda, \mu, t $, which is then maximized over $ c \in (0,2) $ to bound the Hankel determinant.
• Critical point analysis is performed by solving $ K'(c,t) = 0 $, identifying local maxima at $ c = 0^+ $, $ c = 2^- $, or $ c = c_0 = \sqrt{-6M_4/M_3} $, depending on the sign of auxiliary expressions $ M_1, M_2, M_3, M_4 $.
• The maximum value of $ K(c,t) $ is computed in four distinct cases based on the signs of $ M_3 $ and $ M_4 $, leading to piecewise upper bounds.
• The results are validated through special cases, including $ \lambda = \mu $, $ \mu = 1 $, and $ \lambda = \mu = 1 $, which recover known bounds and confirm consistency.

Experimental results

Research questions

• RQ1 What is the sharp upper bound for the second Hankel determinant $ |a_2a_4 - a_3^2| $ in the class $ \mathcal{N}_{\sigma}^{\mu}(\lambda,t) $ of bi-univalent functions defined via Chebyshev polynomials?
• RQ2 How do the parameters $ \lambda $, $ \mu $, and $ t $ influence the magnitude of the second Hankel determinant in this class?
• RQ3 Can the bound be improved over existing results in the literature, particularly those of Mustafa [28], through critical point analysis of the coefficient functional?
• RQ4 What are the explicit expressions for the upper bound in different parameter regimes, especially when the critical point $ c_0 $ lies within $ (0,2) $?

Key findings

• The second Hankel determinant $ |a_2a_4 - a_3^2| $ is bounded above by $ \frac{4t^2}{(2\lambda + \mu)^2} $ when $ M_1 \geq 0 $ and $ M_2 \geq 0 $, corresponding to a maximum at $ c = 0^+ $.
• When $ M_3 \geq 0 $ and $ M_4 \geq 0 $, the bound is $ K(2^{-}, t) = \frac{4t^2}{(2\lambda+1)^2} + \frac{M_3 + 3M_4}{6(\lambda+1)^4(2\lambda+1)^2(3\lambda+1)} $, achieved at $ c = 2^- $.
• For $ M_3 < 0 $ and $ M_4 > 0 $, the maximum occurs at the critical point $ c_0 = \sqrt{-6M_4/M_3} $, yielding $ K(c_0, t) = \frac{4t^2}{(2\lambda+1)^2} - \frac{3M_4^2}{8M_3(\lambda+1)^4(2\lambda+1)^2(3\lambda+1)} $.
• In the case $ \lambda = \mu $, the bound is $ t^2(1 - t^2) $ for $ \frac{1}{2} < t \leq t_{01} \approx 0.603615 $, and $ \frac{t(260t^4 + 84t^3 - 139t^2 - 18t + 9)}{8(18t^3 + 42t^2 - 17t - 9)} $ for $ t_{01} < t < 1 $.
• For the subclass $ \mathcal{S}_{\sigma}^*(t) $, the bound is $ \frac{8t^2}{3} $ when $ \frac{1}{2} < t \leq \frac{7 + \sqrt{401}}{44} \approx 0.6036 $, and $ t^2 + \frac{t(2 + t - 11t^2)^2}{3(22t^2 - 7t - 4)} $ for larger $ t $.
• The derived bounds improve upon the results of Mustafa [28], particularly in the parameter regimes where the critical point $ c_0 $ yields a strictly larger maximum than the endpoint values.