Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Results on Logarithmic Coefficients for the Class of Bounded Turning Functions

Sanju Mandal, Molla Basir Ahamed

TL;DR

This work investigates sharp bounds for logarithmic coefficients and their inverses, as well as second Hankel determinants, within the bean-shaped bounded turning class $\mathcal{BT_\mathfrak{B}}$. Employing differential subordination, Schwarz–Carath\'eodory parametrizations, and explicit extremal constructions, the authors derive precise bounds: $|\gamma_n|\le\tfrac{1}{4(n+1)}$ for $n=1,2,3,4$, $|H_{2,1}(F_f/2)|\le\tfrac{1}{144}$, $|\Gamma_1|\le\tfrac{1}{8}$, $|\Gamma_2|\le\tfrac{1}{12}$, and $|H_{2,1}(F_{f^{-1}}/2)|\le\tfrac{1}{144}$. They also establish a sharp generalized Zalcman inequality $|a_2a_3-a_4|\le\tfrac{1}{8}$ and sharp moduli-difference bounds for both logarithmic and inverse logarithmic coefficients. The results extend the understanding of coefficient problems in univalent and bounded-turning function classes, with explicit extremals confirming sharpness and potential implications for related geometric-function-theory conjectures.

Abstract

It is crucial to explore the sharp bounds of logarithmic coefficients and the Hankel determinant involving logarithmic coefficients as part of coefficient problems in various function classes. Our primary objective in this study is to determine the sharp bounds for logarithmic coefficients as well as logarithmic inverse coefficients of bounded analytic functions associated with a bean-shaped domain in the class $\mathcal{BT_\mathfrak{B}}$. For this class, we also establish the sharp bounds for the second Hankel determinant involving logarithmic coefficients as well as logarithmic inverse coefficients. In addition, we establish sharp bounds for the generalized Zalcman conjecture inequality and the moduli differences of logarithmic coefficients for the class $\mathcal{BT_\mathfrak{B}}$.

Results on Logarithmic Coefficients for the Class of Bounded Turning Functions

TL;DR

This work investigates sharp bounds for logarithmic coefficients and their inverses, as well as second Hankel determinants, within the bean-shaped bounded turning class . Employing differential subordination, Schwarz–Carath\'eodory parametrizations, and explicit extremal constructions, the authors derive precise bounds: for , , , , and . They also establish a sharp generalized Zalcman inequality and sharp moduli-difference bounds for both logarithmic and inverse logarithmic coefficients. The results extend the understanding of coefficient problems in univalent and bounded-turning function classes, with explicit extremals confirming sharpness and potential implications for related geometric-function-theory conjectures.

Abstract

It is crucial to explore the sharp bounds of logarithmic coefficients and the Hankel determinant involving logarithmic coefficients as part of coefficient problems in various function classes. Our primary objective in this study is to determine the sharp bounds for logarithmic coefficients as well as logarithmic inverse coefficients of bounded analytic functions associated with a bean-shaped domain in the class . For this class, we also establish the sharp bounds for the second Hankel determinant involving logarithmic coefficients as well as logarithmic inverse coefficients. In addition, we establish sharp bounds for the generalized Zalcman conjecture inequality and the moduli differences of logarithmic coefficients for the class .

Paper Structure

This paper contains 10 sections, 7 theorems, 124 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Sharp bound of logarithmic coefficients for functions in the class $\mathcal{A}$:
  3. Logarithmic coefficients for the Class $\mathcal{BT_\mathfrak{B}}$:
  4. Sharp bound of $H_{2,1}(F_f/2)$ for the Class $\mathcal{BT_\mathfrak{B}}$:
  5. Sharp bound of logarithmic coefficients for inverse functions in the class $\mathcal{BT_\mathfrak{B}}$
  6. Logarithmic inverse coefficients for the Class $\mathcal{BT_\mathfrak{B}}$:
  7. Sharp bound of $H_{2,1}(F_{f^{-1}}/2)$ for the Class $\mathcal{BT_\mathfrak{B}}$:
  8. Generalized Zalcman conjecture for the Class $\mathcal{BT_\mathfrak{B}}$:
  9. Moduli differences of logarithmic coefficients for $\mathcal{BT_\mathfrak{B}}$
  10. Conclusion

Key Result

Theorem 3.1

Let $f(z)=z+a_2z^2+a_3z^3+\cdots\in\mathcal{BT_\mathfrak{B}}$ and $\gamma_{1}, \gamma_{2}, \gamma_{3}, \gamma_{4}$ are given by eq-2.2. Then we have The inequality is sharp for the following functions:

Figures (7)

  • Figure 1: The image $\mathfrak{B}(\mathbb{D})$ is a bean shaped domain by the function $w=\mathfrak{B}(z)=\sqrt{1+\tanh z}$.
  • Figure 2: Pictorial representation of $f_1(z)=z +\frac{z^2}{4} -\frac{z^3}{24} -\frac{5z^4}{192} +\cdots$.
  • Figure 3: Pictorial representation of $f_2(z)=z +\frac{z^3}{6} -\frac{z^5}{40}-\frac{5z^7}{336}+\cdots$.
  • Figure 4: Pictorial representation of $f_3(z)=z +\frac{z^4}{8} -\frac{z^7}{56}+\cdots$.
  • Figure 5: Pictorial representation of $f_4(z)=z +\frac{z^5}{10} -\frac{z^9}{72}+\cdots$.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 6 more