[Paper Review] Second Hankel determinant of logarithmic coefficients of certain analytic functions

Vasudevarao Allu, Vibhuti Arora|arXiv (Cornell University)|Oct 11, 2021

Analytic and geometric function theory4 citations

TL;DR

This paper establishes sharp upper bounds for the second Hankel determinant of logarithmic coefficients, $ |H_{2,1}(F_f/2)| $, for subclasses of analytic and univalent functions in the unit disk, using parametric representations of Carathéodory functions and optimization over complex parameters. The key result provides explicit sharp bounds for convex and spirallike functions, with numerical values confirmed for specific cases such as $ \lambda = 1/2 $ and $ \lambda = 1 $.

ABSTRACT

We consider a family of all analytic and univalent functions (i.e., one-to-one) in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we obtain the sharp bounds of the second Hankel determinant of Logarithmic coefficients for some subclasses of analytic functions.

Motivation & Objective

• Address the open problem of estimating sharp bounds for the second Hankel determinant of logarithmic coefficients $ H_{2,1}(F_f/2) $ in subclasses of univalent functions.
• Extend previous results on Hankel determinants by focusing on logarithmic coefficients rather than Taylor coefficients.
• Provide a unified framework using parametric representations of functions in the Carathéodory class to derive sharp inequalities.
• Confirm sharpness of bounds through explicit construction of extremal functions.
• Generalize results for convex and spirallike functions by introducing a parameter $ \lambda \in [1/2, 1] $.

Proposed method

• Use the parametric representation of Carathéodory functions $ p \in \mathcal{P} $ with coefficients $ c_1, c_2, c_3 $ expressed in terms of complex parameters $ p_1 \in [0,1] $, $ p_2, p_3 \in \overline{\mathbb{D}} $.
• Express the logarithmic coefficients $ \gamma_n $ in terms of Taylor coefficients $ a_n $, and derive the second Hankel determinant $ H_{2,1}(F_f/2) = \gamma_1\gamma_3 - \gamma_2^2 $ as a quartic form in $ a_2, a_3, a_4 $.
• Apply the transformation $ a_2 = \frac{(2\lambda+1)}{2} p_1 $, $ a_3 = \frac{(2\lambda+1)}{6} ((3+2\lambda)p_1^2 - 1) $, and $ a_4 = \frac{(2\lambda+1)(2\lambda+3)}{24} ((2\lambda+5)p_1^2 - 3)p_1 $ to map the problem into the parameter space.
• Optimize the determinant expression over $ p_1 \in [0,1] $ by analyzing critical points and boundary behavior, using inequalities involving $ |A|, |B|, |C| $ derived from coefficient expressions.
• Establish sharpness by constructing an extremal function $ f_3 $ corresponding to $ p_3(z) = \frac{1 - z^2}{1 - 2s_3 z + z^2} $ with $ s_3 $ defined via the critical point equation.
• Verify that the maximum value is attained at $ s_3 $, confirming the bound is sharp for the given class.

Experimental results

Research questions

• RQ1 What is the sharp upper bound for the second Hankel determinant of logarithmic coefficients $ |H_{2,1}(F_f/2)| $ in the class of convex functions?
• RQ2 How does the bound vary for the more general class of $ \lambda $-spirallike functions with $ \lambda \in [1/2, 1] $?
• RQ3 Can the extremal function achieving the sharp bound be explicitly constructed for these classes?
• RQ4 Is the bound invariant under rotation, and how does this affect the optimization process?
• RQ5 Does the bound derived via parametric representation of Carathéodory functions yield a tighter estimate than previous results?

Key findings

• The sharp upper bound for $ |H_{2,1}(F_f/2)| $ in the class of convex functions ($ \lambda = 1/2 $) is $ 0.030303 $, confirmed as sharp.
• For the class $ \mathcal{C}(-1/2) $, corresponding to $ \lambda = 1 $, the sharp bound is $ 0.070811 $, achieved by an explicit extremal function.
• The bound is derived as $ \frac{(2\lambda+1)^2(12\lambda^2 - 60\lambda - 165)}{576(4\lambda^2 - 12\lambda - 39)} $, valid for $ \lambda \in [1/2, 1] $.
• The extremal function is constructed via $ p_3(z) = \frac{1 - z^2}{1 - 2s_3 z + z^2} $, where $ s_3 $ is the critical point solving $ h'(s_3) = 0 $.
• Numerical verification confirms that $ h(\sqrt{y_2}) = T(\sqrt{y_2}) $, ensuring continuity and maximality at the transition point.
• The bound is sharp because equality is achieved for the constructed function $ f_3 $, with coefficients $ a_2, a_3, a_4 $ explicitly computed from $ s_3 $.