  The well-known Bennett-Hoeffding bound for sums of independent random
variables is refined, by taking into account truncated third moments, and at
that also improved by using, instead of the class of all increasing exponential
functions, the much larger class of all generalized moment functions f such
that f and f" are increasing and convex. It is shown that the resulting bounds
have certain optimality properties. Comparisons with related known bounds are
given. The results can be extended in a standard manner to (the maximal
functions of) (super)martingales.
