# Inequalities: Theory of Majorization and Its Applications

@inproceedings{Marshall1980InequalitiesTO, title={Inequalities: Theory of Majorization and Its Applications}, author={Albert W. Marshall and Ingram Olkin and Barry C. Arnold}, year={1980} }

Although they play a fundamental role in nearly all branches of mathematics, inequalities are usually obtained by ad hoc methods rather than as consequences of some underlying "theory of inequalities." For certain kinds of inequalities, the notion of majorization leads to such a theory that is sometimes extremely useful and powerful for deriving inequalities. Moreover, the derivation of an inequality by methods of majorization is often very helpful both for providing a deeper understanding and…

## 6,338 Citations

Inequalities via Majorization — An Introduction

- Mathematics
- 1983

This paper provides a brief introduction to the theory of majorization and its use in deriving inequalities. Majorization is a preordering of vectors, and inequalities are obtained from the fact that…

More Accurate Majorization Inequalities Obtained Via Superquadraticity and Convexity with Application to Entropies

- Mathematics
- 2021

Different terms such as variability, inequality and dispersion, which occur in various engineering problems and scientific fields, in mathematics are most simply described by the concept of…

General Inequalities For Generalized Convex Functions

- Mathematics
- 2009

It is a fact that, the theory of inequalities, priding on a history of more than two centuries, plays a signi�cant role in almost all �elds of mathematics and in major areas of science. In the…

The optimization for the inequalities of power means

- Mathematics
- 2006

Let be theth power mean of a sequence of positive real numbers, where, and. In this paper, we will state the important background and meaning of the inequality; a necessary and sufficient condition…

Some log-majorizations and an extension of a determinantal inequality

- MathematicsLinear Algebra and its Applications
- 2018

Some results on majorization and their applications

- MathematicsJ. Comput. Appl. Math.
- 2016

Generalizations of Popoviciu's inequality

- Mathematics
- 2008

We establish a general criterion for the validity of inequalities of the following form: A certain convex combination of the values of a convex function at n points and of its value at a weighted…

On a refinement of the majorisation type inequality

- Mathematics
- 2011

Abstract In this note, two mean value theorems are proved by using some recent results by Barnett et al. [N. S. Barnett, P. Cerone, S. S. Dragomir, Majorisation inequalities for Stieltjes integrals,…

Book in Review

- Mathematics
- 1999

Marshall and Olkin’s 1979 book Inequalities: Theory of Majorization and Its Applications has been well received over the last 30 years or so. It has had much impact on the mathematical sciences and…

## References

SHOWING 1-10 OF 12 REFERENCES

Reflection Groups, Generalized Schur Functions, and the Geometry of Majorization

- Mathematics
- 1977

Let G be a closed subgroup of the orthogonal group O(n) acting on Rn. A real-valued function f on Rn is called G-monotone (decreasing) if f(y) > f(x) whenever y ; x, i.e., whenever y e C(x), where…

Inequalitites on the probability content of convex regions for elliptically contoured distributions

- History
- 1972

This work was supported in part by the National Science Foundation Grant No. 17172 at Stanford University, Grants No. 11021, 9593, and 21074 at the University of Minnesota and Grant No. 25911 at the…

The one-sided barrier problem for Gaussian noise

- Mathematics
- 1962

This paper is concerned with the probability, P[T, r(τ)], that a stationary Gaussian process with mean zero and covariance function r(τ) be nonnegative throughout a given interval of duration T.…

Multivariate probability inequalities: an overview, Tech

- Rep. No. 378, Univ. of Minnesota, August
- 1980