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Second Hankel Determinant for Logarithmic Inverse Coefficients of Convex and Starlike Functions

Vasudevarao Allu, Amal Shaji

TL;DR

The paper determines sharp bounds for the second Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ of logarithmic inverse coefficients for convex and starlike functions. It employs a Carathéodory parametrization of coefficient constraints and a sharp bound lemma to optimize over feasible parameter ranges, yielding explicit extremal configurations. Specifically, it proves $|H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{33}$ for convex functions and $|H_{2,1}(F_{f^{-1}}/2)| \le \frac{13}{12}$ for starlike functions, with equalities attained by a chosen Carathéodory extremal and the Koebe function, respectively. These results advance the understanding of logarithmic inverse coefficient behavior and identify exact extremals for classical subclasses of univalent functions.

Abstract

In this paper, we obtain the sharp bounds of the second Hankel determinant of logarithmic inverse coefficients for the starlike and convex functions.

Second Hankel Determinant for Logarithmic Inverse Coefficients of Convex and Starlike Functions

TL;DR

The paper determines sharp bounds for the second Hankel determinant of logarithmic inverse coefficients for convex and starlike functions. It employs a Carathéodory parametrization of coefficient constraints and a sharp bound lemma to optimize over feasible parameter ranges, yielding explicit extremal configurations. Specifically, it proves for convex functions and for starlike functions, with equalities attained by a chosen Carathéodory extremal and the Koebe function, respectively. These results advance the understanding of logarithmic inverse coefficient behavior and identify exact extremals for classical subclasses of univalent functions.

Abstract

In this paper, we obtain the sharp bounds of the second Hankel determinant of logarithmic inverse coefficients for the starlike and convex functions.
Paper Structure (3 sections, 4 theorems, 72 equations)

This paper contains 3 sections, 4 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Main Results

Key Result

Lemma 2.1

If $p \in \mathcal{P}$ is of the form p, then and for some $p_{1}, p_{2}, p_{3} \in \overline{\mathbb{D}}:=\{z \in \mathbb{C}:|z| \leq 1\}$. For $p_{1} \in \mathbb{T}:=\{z \in \mathbb{C}:|z|=1\}$, there is a unique function $p \in \mathcal{P}$ with $c_{1}$ as in c1, namely For $p_{1} \in \mathbb{D}$ and $p_{2} \in \mathbb{T}$, there is a unique function $p \in \mathcal{P}$ with $c_{1}$ and $c_{

Theorems & Definitions (6)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof