  Recently, there has been a number of good deal of research on the Bohr's
phenomenon in various setting including a refined formulation of his classical
version of the inequality. Among them, in \cite{PaulPopeSingh-02-10} the
authors considered the cases in which the above functions have a multiple zero
at the origin. In this article, we present a refined version of Bohr's
inequality for these cases and give a partial answer to a question from
\cite{PaulPopeSingh-02-10} for the revised setting.
