Journal of Inequalities and Applications
Volume 1 (1997), Issue 1, Pages 85-98

Relationship between stochastic inequalities and some classical mathematical inequalities

Y. L. Tong

School of Mathematics, Georgia Institute of Technology, Atlanta 30332-0160, GA, USA

Received 2 May 1996

Copyright © 1997 Y. L. Tong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The notions of association and dependence of random variables, rearrangements, and heterogeneity via majorization ordering have proven to be most useful for deriving stochastic inequalities. In this survey article we first show that these notions are closely related to three basic inequalities in classical mathematical analysis: Chebyshev’s inequality, the Hardy-Littlewood-Pólya rearrangement inequality and Schur functions. We then provide a brief review of some of the recent results in this area. An overall objective is to illustrate that classical mathematical inequalities of this type play a central role in the developments of stochastic inequalities.