Limits of Mellin coefficients and Berezin transform

We consider a bounded radial function f. B. Korenblum and K.E.Zhu give a case where we have equality between the limit near the boundary of the unit disc of the Berezin transform and the limit of the normalized Mellin coefficient when one of them is 0. We previously describe the case where one limit point has modulus infinite norm of f. We also use the mean values of f near 1. The aim of this article is to show that between these two extreme cases, we can have distinct limits. In the same time, we also study the sets of limit points of these quantities.