A note on an area-type functional of Bloch functions

Abstract

Let D be the unit disk centered at the origin in the complex plane. In this paper we consider an extremal problem for an area-type functional in the space B of Bloch functions with the seminorm ||b||_B = sup{(1 − |z|²)|b’(z)|: z ∈ D}. We show that sup{Σ_k=1ⁿ k|b_k|2: ||b||_B ≤ 1} = nB_n², n = 1, 2, 3, 4, 5, where b_k are the Taylor coefficients of b and B_n = sup{|b_n|: ||b||_B ≤ 1}.