Journal of Inequalities and Applications
Volume 5 (2000), Issue 5, Pages 433-446

Some localization theorems using a majorization technique

Monica Bianch and Anna Torriero

Institute of Econometrics and Appfied Mathematics, Catholic University, L. go. A. Gemelli, 1, Milan 1-20123, Italy

Received 3 June 1999; Revised 30 September 1999

Copyright © 2000 Monica Bianch and Anna Torriero. 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.


In this note we localize ordered real numbers through their upper and lower bounds solving a class of nonlinear optimization problems. To this aim, a majorization technique, which involves Schur-convex functions, has been applied and maximum and minimum elements of suitable sets are considered. The bounds we develop can be expressed in terms of the mean and higher centered moments of the number distribution. Meaningful results are obtained for real eigenvalues of a matrix of order n. Finally, numerical examples are provided, showing how former results in the literature can be sometimes improved through those methods.