  In this article, we prove new inequalities between some common probability
metrics. Using these inequalities, we obtain novel local limit theorems for the
magnetization in the Curie-Weiss model at high temperature, the number of
triangles and isolated vertices in Erd\H{o}s-R\'{e}nyi random graphs, as well
as the independence number in a geometric random graph. We also give upper
bounds on the rates of convergence for these local limit theorems and also for
some other probability metrics. Our proofs are based on the Landau-Kolmogorov
inequalities and new smoothing techniques.
