  In this paper we study the behaviour of the Igusa-Todorov functions for Artin
algebras A with finite injective dimension, and Gorenstein algebras as a
particular case. We show that the $\phi$-dimension and $\psi$-dimension are
finite in both cases. Also we prove that monomial, gentle and cluster tilted
algebras have finite $\phi$-dimension and finite $\psi$-dimension.
