[Paper Review] Construction of Toeplitz Matrices whose elements are the coefficients of univalent functions associated with $q$-derivative operator

N. Magesh, Şahsene Altınkaya+1 more|arXiv (Cornell University)|Aug 11, 2017

Analytic and geometric function theory4 references3 citations

TL;DR

This paper constructs coefficient bounds for symmetric Toeplitz determinants of univalent functions in a new subclass $ R(q) $, defined using the $ q $-derivative operator. By relating the $ q $-derivative to functions with positive real part and applying coefficient estimates via Carathéodory functions, the authors derive explicit upper bounds for $ |T_2(2)| $, $ |T_2(3)| $, $ |T_3(2)| $, and $ |T_3(1)| $, with tight estimates depending on $ q \in (0,1) $, generalizing classical results in geometric function theory.

ABSTRACT

In this paper, we find the coefficient bounds using symmetric Toeplitz determinants for the functions belonging to the subclass $R(q)$.

Motivation & Objective

• To investigate symmetric Toeplitz determinants for a new subclass $ R(q) $ of univalent functions defined via the $ q $-derivative operator.
• To extend classical coefficient estimates in geometric function theory by incorporating $ q $-calculus tools.
• To establish explicit upper bounds for $ |T_2(2)| $, $ |T_2(3)| $, $ |T_3(2)| $, and $ |T_3(1)| $ in the class $ R(q) $.
• To generalize results from the classical $ RT $ class (where $ \Re(f'(z)) > 0 $) to the $ q $-analytic setting.

Proposed method

• Define the class $ R(q) $ as analytic functions $ f(z) = z + \sum_{n=2}^\infty a_n z^n $ satisfying $ \Re(D_q f(z)) > 0 $ in the unit disk.
• Use the $ q $-derivative operator $ D_q f(z) = 1 + \sum_{n=2}^\infty [n]_q a_n z^{n-1} $, where $ [n]_q = (1 - q^n)/(1 - q) $.
• Relate $ D_q f(z) $ to a Carathéodory function $ p(z) $ with $ \Re(p(z)) > 0 $, so $ D_q f(z) = p(z) $.
• Express coefficients $ a_n $ in terms of $ p_k $ via $ a_n = p_{n-1} / [n]_q $, and apply coefficient bounds from Lemma 2.2 on $ p_k $.
• Apply the triangle inequality and calculus-based maximization over $ |x| \in [0,1] $ and $ p \in [0,2] $ to derive upper bounds for the Toeplitz determinants.
• Use algebraic simplification and substitution to express the final bounds in terms of $ q $, such as $ \frac{4q^2(q^2 + 2q + 2)}{(1 + 2q + 2q^2 + q^3)^2} $.

Experimental results

Research questions

• RQ1 What are the sharp upper bounds for the symmetric Toeplitz determinants $ |T_2(2)| $, $ |T_2(3)| $, $ |T_3(2)| $, and $ |T_3(1)| $ in the class $ R(q) $?
• RQ2 How does the $ q $-derivative operator modify the coefficient estimates compared to the classical $ RT $ class?
• RQ3 Can the bounds for Toeplitz determinants in $ R(q) $ be expressed explicitly in terms of $ q \in (0,1) $?
• RQ4 What is the role of the Carathéodory function parameterization in deriving these bounds?
• RQ5 How do the bounds behave as $ q \to 1^- $, and do they recover known results from the classical case?

Key findings

• The bound for $ |T_2(2)| $ is $ \frac{4q^2(q^2 + 2q + 2)}{(1 + 2q + 2q^2 + q^3)^2} $, achieved when $ p = 2 $ and $ |x| = 1 $.
• The bound for $ |T_2(3)| $ is $ \frac{16q^2}{(1+q+q^2+q^3)(1+q+q^2)^2} $, derived via maximization of a function in $ |x| \in [0,1] $.
• The bound for $ |T_3(2)| $ is $ \frac{16q^2}{(1+q+q^2+q^3)(1+q+q^2)^2} $, obtained by combining estimates for $ |a_2 - a_4| $ and $ |a_2^2 - 2a_3^2 + a_2 a_4| $.
• The bound for $ |T_3(1)| $ is $ 1 + \frac{4}{(1+q+q^2)^2} $, with maximum achieved at $ p = 2 $.
• All bounds are sharp and depend explicitly on $ q $, with convergence to classical results as $ q \to 1^- $.
• The derivations rely on Lemma 2.2 to express $ p_2 $ and $ p_3 $ in terms of $ p_1 $, $ x $, and $ z $, and use calculus to maximize over $ |x| \in [0,1] $.