Correlations equalities and some upper bounds for the coupling constant implying area decay of Wilson loop for $Z_3$ lattice gauge theories

Correlation identities are obtained for $Z_3$ lattice gauge theory where the bonds of the plaquettes are decorated by generalized three-state Ising variables. Making use of correlation inequalities we obtain the area decay of the Wilson loop observable in a range of the coupling parameter larger than those obtained from mean field theory considerations.