Certain statistical problems in life testing and reliability lead to the consideration of minimizing and maximizing certain integrals of the form $\int^\infty_0 \phi(x, \bar F(x)) dx,\quad\bar F(x) = 1 - F(x)$, under the assumption that the distribution function $F$ has increasing hazard rate and that one or two moments are given. The minimizing (maximizing) distributions for the special cases considered are members of the extremal class considered by A. Marshall and the author . Bounds on the expected values of the minimum and maximum order statistics as well as the operating characteristics of replacement policies based on age are obtained under these assumptions.
"Bounds on Integrals with Applications to Reliability Problems." Ann. Math. Statist. 36 (2) 565 - 574, April, 1965. https://doi.org/10.1214/aoms/1177700168