  We analyse the cluster luminosity functions (CLFs) of the youngest star
clusters in three galaxies exhibiting prominent circumnuclear starburst rings.
We focus specifically on NGC 1512 and NGC 6951, for which we have access to
H$\alpha$ data that allow us to unambiguously identify the youngest sample
clusters. To place our results on a firm statistical footing, we first explore
in detail a number of important technical issues affecting the process from
converting the observational data into the spectral-energy distributions of the
objects in our final catalogues. The CLFs of the young clusters in both
galaxies exhibit approximate power-law behaviour down to the 90 per cent
observational completeness limits, thus showing that star cluster formation in
the violent environments of starburst rings appears to proceed similarly as
that elsewhere in the local Universe. We discuss this result in the context of
the density of the interstellar medium in our starburst-ring galaxies.
