Let Xa j, be nonnegative random variables with the property that Xa, b < xa, c + x'c.b for all 0 < a < c < b < T, where T >0 is fixed. We define Maj,:= sup{Aa, c: a < c < b} and establish bounds for EMya b and Eexp{XMa A) in terms of assumed bounds for EXya b and Eexp(AXa b), respectively, where y >1 and X runs through an interval (Ao, oo) with fixed Xq >0. These bounds explicitly involve a nonnegative function g{a, b) assumed to be quasi-superadditive with an index Q, which means that g{a, c) + g{c, b) < Qg{a, b) for all 0 < a < c < b < T, where 1 < Q < 2 is fixed. Maximal inequalities obtained in this way can be applied to stochastic processes exhibiting long-range dependence. Among others, these applications may include certain self-similar processes such as fractional Brownian motion, stochastic processes occurring in linear time-series models, etc. © 1994 American Mathematical Society.