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Sharp Estimates of Logarithmic Coefficients for a Certain Class of Starlike Functions

Molla Basir Ahamed, Sanju Mandal

Abstract

In this article, we investigate the extremal properties of logarithmic coefficients for the class $\mathcal{S}_{ch}^*$ of starlike functions associated with the hyperbolic cosine function. We establish the sharp upper bounds for the initial logarithmic coefficients $γ_n$ for $n=1, 2, 3$, and determine the precise bound for the second Hankel determinant $H_{2,1}(F_f/2)$ within this class. Furthermore, we extend our analysis to the inverse functions, deriving sharp estimates for the logarithmic inverse coefficients and the corresponding second Hankel determinant $|H_{2,1}(F_{f^{-1}}/2)|$. Additionally, we provide sharp bounds for the moduli differences of both logarithmic and inverse logarithmic coefficients. The sharpness of all obtained inequalities is verified through the construction of specific extremal functions.

Sharp Estimates of Logarithmic Coefficients for a Certain Class of Starlike Functions

Abstract

In this article, we investigate the extremal properties of logarithmic coefficients for the class of starlike functions associated with the hyperbolic cosine function. We establish the sharp upper bounds for the initial logarithmic coefficients for , and determine the precise bound for the second Hankel determinant within this class. Furthermore, we extend our analysis to the inverse functions, deriving sharp estimates for the logarithmic inverse coefficients and the corresponding second Hankel determinant . Additionally, we provide sharp bounds for the moduli differences of both logarithmic and inverse logarithmic coefficients. The sharpness of all obtained inequalities is verified through the construction of specific extremal functions.
Paper Structure (6 sections, 11 theorems, 132 equations, 1 figure)

This paper contains 6 sections, 11 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. New problem formulations for the class $\mathcal{S}^*_{ch}$
  3. Sharp bound of logarithmic coefficients for functions in the class $\mathcal{S}^*_{ch}$:
  4. Sharp bound of logarithmic coefficients for inverse functions in the class $\mathcal{S}^*_{ch}$
  5. Moduli differences of logarithmic coefficients and inverse counterpart for functions in the class $\mathcal{S}^*_{ch}$
  6. Concluding remarks

Key Result

Lemma 2.1

Libera-Zlotkiewicz-PAMS-1982Libera-Zlotkiewicz-PAMS-1983 If $p\in\mathcal{P}$ is of the form eq-2.4 with $c_1\geq 0$, then and for some $\tau_1\in[0,1]$ and $\tau_2,\tau_3\in\overline{\mathbb{D}}:= \{z\in\mathbb{C}:|z|\leq 1\}$. For $\tau_1\in\mathbb{T}:=\{z\in\mathbb{C}:|z|=1\}$, there is a unique function $p\in\mathcal{P}$ with $c_1$ as in eq-2.5, namely For $\tau_1\in\mathbb{D}$ and $\tau_2\

Figures (1)

  • Figure 1: The geometrical representation of $z+\cosh (z)$.

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['Th-22.11']}
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['Th-2.1']}
  • ...and 11 more