  A general reverse Pinsker's inequality is derived to give an upper bound on
f-divergences in terms of total variational distance when two distributions are
close measured under our proposed generalized local information geometry
framework. In addition, relationships between two f-divergences equipped with
functions that are third order differentiable are established in terms of the
lower and upper bounds of their ratio, when the underlying distributions are
within a generalized quasi-$\varepsilon$-neighborhood.
