On a powered Bohr inequality

Abstract

The object of this paper is to study the powered Bohr radius ${\rho }_p$, $p\in (1,2)$, of analytic functions $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$ 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 $M_p^f(r)\le 1$ for $r\le r_p$ 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.

11 pages; To appear in Annales Academiae Scientiarum Fennicae Mathematica