The Range of the Berezin Transform

Resumo

We give a description of all functions of the form \( \sum \limits_{i=1}^N{f}_i{\overline{g}}_i \) in the range of the Berezin transform, where f_i and g_i, 1 ≤ i ≤ N, are holomorphic functions on the unit disk ????, which generalizes the corresponding result of Ahern. We also give a very simple proof of the results of Ahern and the results due to Čučković and Li who deal with the special case N = 2 and g₂ = zⁿ.