  In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30],
several inequalities for tail probabilities of sums M_n=X_1+... +X_n of bounded
independent random variables X_j were proved. These inequalities had a
considerable impact on the development of probability and statistics, and
remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ.
Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two
theorems. By similar factors, a third theorem was refined by Pinelis [Progress
in Probability 43 (1998) 257-314] and refined (and extended) by me. In this
article, I introduce a new type of inequality.
  Namely, I show that P{M_n\geq x}\leq cP{S_n\geq x}, where c is an absolute
constant and S_n=\epsilon_1+... +\epsilon_n is a sum of independent identically
distributed Bernoulli random variables (a random variable is called Bernoulli
if it assumes at most two values). The inequality holds for those x\in R where
the survival function x\mapsto P{S_n\geq x} has a jump down. For the remaining
x the inequality still holds provided that the function between the adjacent
jump points is interpolated linearly or \log-linearly. If it is necessary, to
estimate P{S_n\geq x} special bounds can be used for binomial probabilities.
The results extend to martingales with bounded differences. It is apparent that
Theorem 1.1 of this article is the most important.
