Schur-convex functions of the 2nd order on ℝⁿ

Revyakov, M.

doi:10.1090/spmj/1627

Schur-convex functions of the 2nd order on $\mathbb {R}^n$
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by M. Revyakov
Translated by: M. Revyakov

St. Petersburg Math. J. 31 (2020), 887-902

DOI: https://doi.org/10.1090/spmj/1627

Published electronically: September 3, 2020

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Abstract:

In the author’s earlier paper [Revyakov M., J. Multivariate Anal. 116 (2013) 25–34] concerning mathematical statistics, a need arose to employ functions called “Schur-convex functions of the 2nd order with respect to two variables”.

In the present paper, the class of Schur-convex functions of the 2nd order in $n$ variables is introduced. Necessary and sufficient conditions (in the form of analogs of the Sylvester criterion) are established for a function to belong to this class. Examples are given of using Schur-convex functions of the 2nd order for achieving maximal system reliability on the set of all possible allocations of elements into its subsystems.

References

B. Bernstein and R. A. Toupin, Some properties of the Hessian matrix of a strictly convex function, J. Reine Angew. Math. 210 (1962), 65–72. MR 145024
Emad El-Neweihi, Frank Proschan, and Jayaram Sethuraman, Optimal assembly of systems using Schur functions and majorization, Naval Res. Logist. 34 (1987), no. 5, 705–712. MR 906431, DOI 10.1002/1520-6750(198710)34:5<705::AID-NAV3220340510>3.0.CO;2-P
F. R. Gantmacher, The theory of matrices. Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998. Translated from the Russian by K. A. Hirsch; Reprint of the 1959 translation. MR 1657129
Peter M. Gruber, Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
Albert W. Marshall, Ingram Olkin, and Barry C. Arnold, Inequalities: theory of majorization and its applications, 2nd ed., Springer Series in Statistics, Springer, New York, 2011. MR 2759813, DOI 10.1007/978-0-387-68276-1
Parametric equation, Encyclopedia of Mathematics. Vol. 4, Soviet Encycl., Moscow, 1984, 221–222. (Russian)
V. Rajendra Prasad, K. P. K. Nair, and Y. P. Aneja, Optimal assignment of components to parallel-series and series-parallel systems, Oper. Res. 39 (1991), no. 3, 407–414. MR 1111628, DOI 10.1287/opre.39.3.407
Mikhail Revyakov, Component allocation for a distributed system: reliability maximization, J. Appl. Probab. 30 (1993), no. 2, 471–477. MR 1212680, DOI 10.1017/s0021900200117498
Mikhail Revyakov, Schur-convexity of 2nd order, certain subclass of multivariate arrangement increasing functions with applications in statistics, J. Multivariate Anal. 116 (2013), 25–34. MR 3049888, DOI 10.1016/j.jmva.2012.11.013