Limits of Mellin coefficients and Berezin transform

TL;DR

This paper investigates the relationship between the limits of Berezin transforms and Mellin coefficients for bounded radial functions, revealing that intermediate cases can have distinct limits and analyzing their limit point sets.

Contribution

It extends previous work by showing that between known extreme cases, the limits of Berezin transform and Mellin coefficients can differ, and studies their limit point sets.

Findings

Limits of Berezin transform and Mellin coefficients can differ in intermediate cases.

The sets of limit points for these quantities are characterized.

Previous extreme cases are expanded upon with new examples.

Abstract

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.