On a powered Bohr inequality

The object of this paper is to study the powered Bohr radius ρ p , p ∈ (1, 2), of analytic functions f(z) =Σ k=0∞ a k z k defined on the unit disk |z| < 1 and such that |f(z)| < 1 for |z| < 1. More precisely, if M pf (r) =Σ k=0∞ |a k | p r k , then we show that M pf (r) ≤ 1 for r ≤ r p where r ρ 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.