  This paper explores some applications of a two-moment inequality for the
integral of the $r$-th power of a function, where $0 < r< 1$. The first
contribution is an upper bound on the R\'{e}nyi entropy of a random vector in
terms of the two different moments. When one of the moments is the zeroth
moment, these bounds recover previous results based on maximum entropy
distributions under a single moment constraint. More generally, evaluation of
the bound with two carefully chosen nonzero moments can lead to significant
improvements with a modest increase in complexity. The second contribution is a
method for upper bounding mutual information in terms of certain integrals with
respect to the variance of the conditional density. The bounds have a number of
useful properties arising from the connection with variance decompositions.
