Exact upper bounds on the generalized moments E ƒ (Sn) of sums Sn of independent nonnegative random variables Xi for certain classes Ƒ of nonincreasing functions ƒ are given in terms of (the sums of) the first two moments of the Xi’s. These bounds are of the form Eƒ(η), where the random variable η is either binomial or Poisson depending on whether n is fixed or not. The classes Ƒ contain, and are much wider than, the class of all decreasing exponential functions. As corollaries of these results, optimal in a certain sense upper bounds on the left-tail probabilities P(Sn ≤ x) are presented, for any real x. In fact, more general settings than the ones described above are considered. Exact upper bounds on the exponential moments Eexp{hSn} for h < 0, as well as the corresponding exponential bounds on the left-tail probabilities, were previously obtained by Pinelis and Utev. It is shown that the new bounds on the tails are substantially better.