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Coefficient estimates and Bohr phenomenon for analytic functions involving semigroup generator

Molla Basir Ahamed, Sanju Mandal

Abstract

This article investigates the Bohr phenomenon and sharp coefficient problems for the class $\mathcal{A}_β$, a subclass of analytic self-maps of the unit disk with the holomorphic generators of one-parameter continuous semigroups. By integrating concepts from complex dynamics and geometric function theory, we derive sharp improvements to the classical Bohr radius by incorporating multiple Schwarz functions and certain functional expressions. We establish generalized versions of the Bohr and Bohr-Rogosinski inequalities and determine the best possible radii for these refinements. Furthermore, we provide a sharp solution to the classical Fekete-Szegö problem for the class $\mathcal{A}_β$ by obtaining sharp bounds for the functional $|a_3 - μa_2^2|$ for all real values of $μ$. Additionally, we derive sharp inequalities for the moduli of differences of logarithmic coefficients for both the functions and their inverses in this class.

Coefficient estimates and Bohr phenomenon for analytic functions involving semigroup generator

Abstract

This article investigates the Bohr phenomenon and sharp coefficient problems for the class , a subclass of analytic self-maps of the unit disk with the holomorphic generators of one-parameter continuous semigroups. By integrating concepts from complex dynamics and geometric function theory, we derive sharp improvements to the classical Bohr radius by incorporating multiple Schwarz functions and certain functional expressions. We establish generalized versions of the Bohr and Bohr-Rogosinski inequalities and determine the best possible radii for these refinements. Furthermore, we provide a sharp solution to the classical Fekete-Szegö problem for the class by obtaining sharp bounds for the functional for all real values of . Additionally, we derive sharp inequalities for the moduli of differences of logarithmic coefficients for both the functions and their inverses in this class.

Paper Structure

This paper contains 4 sections, 12 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Refined Bohr and Bohr-Rogosinski phenomena for the class $\mathcal{A}_{\beta}$
  3. The sharp Fekete-Szegö inequality for functions in the class $\mathcal{A}_{\beta}$
  4. Moduli differences of logarithmic coefficients for functions and inverse functions in the class $\mathcal{A}_{\beta}$

Key Result

Theorem 2.1

If $f\in \mathcal{A}_{\beta}$ is of the form Eq-2.1 and $F : [0, \infty)\to [0, \infty)$ is an monotone increasing function with $F(0)=0$, then for $m\in\mathbb{N}$, $p>0$, the inequality holds for $|z|\leq R^m_{\beta, p}$, where $R^m_{\beta, p}$ is the smallest positive root in $(0, 1)$ of The radius $R^m_{\beta, p}$ is sharp.

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Corollary 2.1
  • Remark 2.3
  • proof : Proof of Theorem \ref{['Th-2.1']}
  • Theorem 2.2
  • Corollary 2.2
  • ...and 12 more