Improved Bohr-type Inequalities for the Cesaro Operator

TL;DR

This paper establishes sharper Bohr-type inequalities for the Cesàro operator on bounded analytic functions in the unit disk, enhancing previous bounds with new analytical techniques.

Contribution

It introduces improved bounds for Bohr-type inequalities specifically tailored for the Cesàro operator acting on bounded analytic functions.

Findings

Derived sharp improved Bohr-type inequalities for the Cesàro operator.

Utilized substitution principles involving the Cesàro operator and Schwarz functions.

Enhanced the bounds for analytic functions in the unit disk.

Abstract

In this paper, we derive the sharp improved versions of Bohr-type inequalities for the Ces\'aro 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 Ces\'aro operator associated with a bounded analytic function defined on $\D$ and its derivative, as well as for the Schwarz function.