  Rindler positivity is a property that holds in any relativistic Quantum Field
Theory and implies an infinite set of inequalities involving the exponential of
the R\'enyi mutual information $I_n(A_i,\bar{A}_j)$ between $A_i$ and
$\bar{A}_j$, where $A_i$ is a spacelike region in the right Rindler wedge and
$\bar{A}_j$ is the wedge reflection of $A_j$. We explore these inequalities in
order to get local inequalities for $I_n(A,\bar{A})$ as a function of the
distance between $A$ and its mirror region $\bar{A}$. We show that the
assumption, based on the cluster property of the vacuum, that $I_n$ goes to
zero when the distance goes to infinity, implies the more stringent and simple
condition that $F_n\equiv{e}^{(n-1)I_n}$ should be a completely monotonic
function of the distance, meaning that all the even (odd) derivatives are
non-negative (non-positive). In the case of a CFT in 1+1 dimensions, we show
that conformal invariance implies stronger conditions, including a sort of
monotonicity of the R\'enyi mutual information for pairs of intervals. An
application of these inequalities to obtain constraints for the OPE
coefficients of the $4-$point function of certain twist operators is also
discussed.
