Correlation inequalities and monotonicity properties of the Ruelle operator

In this paper, we provide sufficient conditions for the validity of the FKG Inequality, on Thermodynamic Formalism setting, for a class of eigenmeasures of the dual of the Ruelle operator. We use this correlation inequality to study the maximal eigenvalue problem for the Ruelle operator associated to low regular potentials. As an application, we obtain explicit upper bounds for the main eigenvalue (consequently, for the pressure) of the Ruelle operator associated to Ising models with a power law decay interaction energy.