<p>This paper aims to establish refined versions of the Bohr phenomenon for various classes of analytic functions. Specifically, we consider functions subordinate to close-to-convex functions, starlike functions of order alpha, functions whose derivatives have a positive real part and functions that are convex in the direction of the imaginary axis. A key aspect of our approach involves analyzing the distance between the image of the origin and the boundary of the image of the unit disk under these functions, which plays a crucial role in deriving the improved Bohr-type inequalities.</p>