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A uniform estimate for rate functions in large deviations

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