A uniform estimate for rate functions in large deviations

Given H\"older continuous functions $f$ and $\psi$ on a sub-shift of finite type $\Sigma_A^{+}$ such that $\psi$ is not cohomologous to a constant, the classical large deviation principle holds (\cite{OP}, \cite{Kif}, \cite{Y}) with a rate function $I_\psi\geq 0$ such that $I_\psi (p) = 0$ iff $p = \int \psi \, d \mu$, where $\mu = \mu_f$ is the equilibrium state of $f$. In this paper we derive a uniform estimate from below for $I_\psi$ for $p$ outside an interval containing $\tilde{\psi} = \int \psi \, d\mu$, which depends only on the sub-shift, the function $f$, the norm $|\psi|_\infty$, the H\"older constant of $\psi$ and the integral $\tilde{\psi}$. Similar results can be derived in the same way e.g. for Axiom A diffeomorphisms on basic sets.