  We show that the Bernstein-Hoeffding method can be employed to a larger class
of generalized moments. This class includes the exponential moments whose
properties play a key role in the proof of a well-known inequality of Wassily
Hoeffding, for sums of independent and bounded random variables whose mean is
assumed to be known. As a result we can generalise and improve upon this
inequality. We show that Hoeffding's bound is optimal in a broader sense. Our
approach allows to obtain "missing" factors in Hoeffding's inequality whose
existence is motivated by the central limit theorem. The later result is a
rather weaker version of a theorem that is due to Michel Talagrand. Using ideas
from the theory of Bernstein polynomials, we show that the Bernstein-Hoeffding
method can be adapted to case in which one has information on higher moments of
the random variables. Moreover, we consider the performance of the method under
additional information on the conditional distribution of the random variables
and, finally, we show that the method reduces to Markov's inequality when
employed to non-negative and unbounded random variables.
