Probability and expectation inequalities

Summary This paper introduces a mathematical framework within which a wide variety of known and new inequalities can be viewed from a common perspective. Probability and expectation inequalities of the following types are considered: (a)P(ZA) P(ZA) for some class of setsA, (b)k(Z)k(Z) for some class of functionsk, and (c)l(Z)k(Z) for some class of pairs of functionsl andk. It is shown, sometimes using explicit constructions ofZ andZ, that, in several cases, (a) (b) (c); included here are cases of normal and elliptically contoured distributions. A case where (a) (b) (c) is studied and is expressed in terms ofn monotone functions for (some of) which integral representations are obtained. Also, necessary and sufficient conditions for (c) are given.Research supported by the Air Force Office of Scientific Research under Grants AFOSR-75-2796 and AFOSR-80-0080Research supported by the National Science Foundation under Grants MCS78-01240 and MCS81-00748