Improved Bohr-type Inequalities for the Cesaro Operator

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 $\text{D}=\{z\in \text{C}:\mid z\mid <1\}$. 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 $\text{D}$ and its derivative, as well as for the Schwarz function.