ON A POWERED BOHR INEQUALITY

The object of this paper is to study the powered Bohr radius rho(p), p is an element of(1, 2), of analytic functions f(z) = Sigma(infinity)(k=0) a(k)z(k) defined on the unit disk vertical bar z vertical bar < 1 and such that vertical bar f(z)vertical bar < 1 for vertical bar z vertical bar < 1. More precisely, if M-p(f) (r) = Sigma(infinity)(k=0) vertical bar a(k)vertical bar(p)r(k), then we show that M-p(f)(r) <= 1 for r <= 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 vertical bar z vertical bar < 1. Finally, we conclude by stating the Bohr radius for the class of Bieberbach-Eilenberg functions.