Higher order optimal approximation of Csiszar's f-divergence

Here are established various sharp and nearly optimal probabilistic inequalities giving the high order approximation of Csiszar's f-divergence between two probability measures, which is the most essential and general tool for their comparison. The above are done through Taylor's formula, generalized Taylor-Widder's formula, an alternative recent expansion formula. Based on these we give many representation formulae of Csiszar's distance, then we estimate in all directions their remainders by using either the norms approach or the modulus of continuity way. Most of the last probabilistic estimates are sharp or nearly sharp, attained by basic simple functions.