On a powered Bohr inequality

The object of this paper is to study the powered Bohr radius ${\rho }_p$, $p \in (1,2)$,ofanalyticfunctions$f(z)=\sum_{k=0}^{\infty} a_kz^k$andsuchthat$|f(z)|<1$definedontheunitdisk$|z|<1$\mathrm{.}Moreprecisely,if$M_p^f ®=\sum_{k=0}^\infty |a_k|^p r^k$,thenweshowthat$M_p^f ®\leq 1$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 $\mathrm{\mid }z\mathrm{\mid }<1$. Finally, we conclude by stating the Bohr radius for the class of Bieberbach-Eilenberg functions.