In this paper, we study improved and refined versions of the classical Bohr inequality for the class \(\mathcal {B}\) , which consists of analytic self-mappings defined on the unit disk \(\mathbb {D}\) . First, we improve the Bohr inequality for the class \(\mathcal {B}\) of analytic self-maps by incorporating the area measurements of sub-disks \(\mathbb {D}_r\) in \(\mathbb {D}\) . Next, we establish a sharp inequality with a 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 \(|f^{(k)}(z)|/k!\) . All the results are proved to be sharp.
