  The object of this paper is to study the powered Bohr radius $\rho_p$, $p \in
(1,2)$, of analytic functions $f(z)=\sum_{k=0}^{\infty} a_kz^k$ and such that
$|f(z)|<1$ defined on the unit disk $|z|<1$. More precisely, if $M_p^f
(r)=\sum_{k=0}^\infty |a_k|^p r^k$, then we show that $M_p^f (r)\leq 1$ for $r
\leq r_p$ where $r_\rho$ is the powered Bohr radius for conformal automorphisms
of the unit disk. This answers the open problem posed by Djakov and Ramanujan
in 2000. A couple of other consequences of our approach is also stated,
including an asymptotically sharp form of one of the results of Djakov and
Ramanujan. In addition, we consider a similar problem for sense-preserving
harmonic mappings in $|z|<1$. Finally, we conclude by stating the Bohr radius
for the class of Bieberbach-Eilenberg functions.
