On a powered Bohr inequality

TL;DR

This paper investigates the powered Bohr radius for analytic functions within the unit disk, resolving an open problem, and extends the analysis to harmonic mappings and Bieberbach-Eilenberg functions.

Contribution

It determines the powered Bohr radius for certain analytic functions, solving an open problem, and explores related harmonic and Bieberbach-Eilenberg function classes.

Findings

Established the powered Bohr radius for $p eq 1,2$

Provided asymptotically sharp bounds for the Bohr radius

Extended results to harmonic mappings and Bieberbach-Eilenberg functions

Abstract

The object of this paper is to study the powered Bohr radius $\rho_p$, $p \in (1,2)$, of analytic functions $f(z)=\sum_{k=0}^{\infty} a_kz^k$ 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.