In this article, we prove several refined versions of the classical Bohr inequality for the class of analytic self-mappings on the unit disk
$ \mathbb {D} $
, class of analytic functions
$ f $
defined on
$ \mathbb {D} $
such that
$\mathrm {Re}\left (f(z)\right )<1 $
, and class of subordination to a function g in
$ \mathbb {D} $
. Consequently, the main results of this article are established as certainly improved versions of several existing results. All the results are proved to be sharp.