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Improved Bohr-type Inequalities for the Cesaro Operator

Vasudevarao Allu, Raju Biswas, Rajib Mandal

Abstract

In this paper, we derive the sharp improved versions of Bohr-type inequalities for the Cesáro operator acting on the class of bounded analytic functions defined on the unit disk $\D=\left\{z\in\C:\left|z\right|<1\right\}$. In order to achieve these results, we utilize the principle of substituting the initial coefficients of the majorant series with the absolute values of the Cesáro operator associated with a bounded analytic function defined on $\D$ and its derivative, as well as for the Schwarz function.

Improved Bohr-type Inequalities for the Cesaro Operator

Abstract

In this paper, we derive the sharp improved versions of Bohr-type inequalities for the Cesáro operator acting on the class of bounded analytic functions defined on the unit disk . In order to achieve these results, we utilize the principle of substituting the initial coefficients of the majorant series with the absolute values of the Cesáro operator associated with a bounded analytic function defined on and its derivative, as well as for the Schwarz function.

Paper Structure

This paper contains 3 sections, 4 theorems, 98 equations, 4 figures.

Table of Contents

  1. Introduction and Preliminaries
  2. Main results and their proofs
  3. Declarations:

Key Result

Lemma 1.1

201 Suppose $f$ is analytic in $\mathbb{D}$ with $|f(z)|\leq1$, then

Figures (4)

  • Figure 1: The graph of $B_2(r)$ in $(0,1)$
  • Figure 2: The graphs of $B_3(r)$ and $B_4(r)$ in $(0,1)$
  • Figure 3: The graph of $B_2'(r)$ in $(0,1)$
  • Figure 4: The graph of the polynomial $F(r)$ in $[0,1]$

Theorems & Definitions (6)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof