  We study the critical behaviour of the $q$-state Potts model on an
uncorrelated scale-free network having a power-law node degree distribution
with a decay exponent $\lambda$. Previous data show that the phase diagram of
the model in the $q,\lambda$ plane in the second order phase transition regime
contains three regions, each being characterized by a different set of critical
exponents. In this paper we complete these results by finding analytic
expressions for the scaling functions and critical amplitude ratios in the
above mentioned regions. Similar to the previously found critical exponents,
the scaling functions and amplitude ratios appear to be $\lambda$-dependent. In
this way, we give a comprehensive description of the critical behaviour in a
new universality class.
