A recent continuous family of multiplicity functions on local rings was
introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz
multiplicities. The obvious goal is to use this as a tool for deforming results
from one to the other. The values in this family which do not match these
classic variants however are not known yet to be well-behaved. This article
explores lower bounds for these intermediate multiplicities as well as gives
evidence for analogies of the Watanabe-Yoshida minimality conjectures for
unmixed singular rings.Comment: 10 page
