Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Revisiting Bohr Inequalities with Analytic and Harmonic Mappings on unit disk

Molla Basir Ahamed, Partha Pratim Roy

Abstract

In this paper, we study some improved and refined versions of the classical Bohr inequality applicable to the class $\mathcal{B}$, which consists of self-analytic mappings defined on the unit disk $\mathbb{D}$. First, we improve the Bohr inequality for the class $\mathcal{B}$ of analytic self-maps, incorporating the area measurements of sub-disks $\mathbb{D}_r$ of $\mathbb{D}$. Secondly, we establish a sharp inequality with suitable setting as an improved version of the classic Bohr inequality. Then we obtain a sharp refined Bohr inequality in which the coefficients $|a_k|$ $(k=0, 1, 2, 3)$ in the majorant series $M_f(r)$ of $f$ are replaced by $|f^{(k)}(z)|/k!$. Finally, for a certain class $\mathcal{P}^0_{\mathcal{H}}(M)$ of harmonic mappings of the form $f=h+\overline{g}$, we generalize the Bohr inequality incorporating a sequence $\{\varphi_n(r)\}_{n=0}^{\infty}$ of continuous functions of $r$ in $[0, 1)$ and give some applications.

Revisiting Bohr Inequalities with Analytic and Harmonic Mappings on unit disk

Abstract

In this paper, we study some improved and refined versions of the classical Bohr inequality applicable to the class , which consists of self-analytic mappings defined on the unit disk . First, we improve the Bohr inequality for the class of analytic self-maps, incorporating the area measurements of sub-disks of . Secondly, we establish a sharp inequality with suitable setting as an improved version of the classic Bohr inequality. Then we obtain a sharp refined Bohr inequality in which the coefficients in the majorant series of are replaced by . Finally, for a certain class of harmonic mappings of the form , we generalize the Bohr inequality incorporating a sequence of continuous functions of in and give some applications.
Paper Structure (9 sections, 26 theorems, 89 equations, 1 table)

This paper contains 9 sections, 26 theorems, 89 equations, 1 table.

Table of Contents

  1. Introduction
  2. Bohr inequalities for the class $\mathcal{B}$ of bounded analytic functions on unit disk $\mathbb{D}$
  3. Basic Notations
  4. Sharp refined Bohr inequality for the class $\mathcal{B}$
  5. Improved Bohr inequality for the class $\mathcal{B}$
  6. Key Lemmas, Proof of Theorems \ref{['Thmm-2.3']} and \ref{['th-2.4']}
  7. Generalization of the Bohr inequality for certain classes of harmonic mappings and its applications
  8. Application of Theorem \ref{['th-7.3']}
  9. Proof of Theorem \ref{['th-7.3']}

Key Result

Theorem 2.1

Kayu-Kham-Ponnu-2021-JMAA Suppose that $f(z)=\sum_{n=0}^{\infty}a_nz^n\in\mathcal{B}$. Then where $R_N$ is the positive root of the equation $2(1+r)r^N-(1-r)^2=0$. The radius $R_N$ is the best possible. Moreover, where $R^{\prime}_N$ is the positive root of the equation $(1+r)r^N-(1-r)^2=0$. The radius $R^{\prime}_N$ is the best possible.

Theorems & Definitions (34)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5: Ahamed-CMFT-2022
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 24 more