Weighted p-majorization and Stable vector

The notion of majorization has been introduced while studying the topic such as wealth distribution, inequalities etc. In the early part of the twentieth century, Lorenz introduced a curve, that describes the wealth (or income) distribution of a population. Let be denote the income profile of a population of size n. Then the line joining origin and the points where be the decreasing rearrangement of the components of $\alpha$, is called Lorenz curve of . If total income or wealth of the population is uniformly distributed among the population then the Lorenz curve is a straight line, otherwise, the curve is convex. Let and be denote the income of a population of size n, in day 1 and day 2 respectively. If x is majorized by y, then Lorenz curve of x is closer to the straight line of uniform distribution than the Lorenz curve of y. This tech- nique is used in many different [6, 7] such as describing inequality among the size of individuals in ecology, in studies of biodiversity, business modeling, etc. The Lorenz curve also provides a different tool for estimating the distributional dimensions of energy consumption. Many authors generalized and studied the majorization by introducing different parameters [1, 2, 5]. In 1947, by introducing weighted p-majorization for a pair of vectors that are in a similar order (either increasing or decreasing) Fuchs [3], proved an equivalent condition for weighted p majorization using continuous convex functions. In 1997, Pˇecar´c and Abramovich [4] discussed an analogs of Fuchs result in which order of one of vector in the pair can be relaxed. In general the wealth (or income) profiles of individuals in a population may not be in a similar order. This motivates us to define majorization for a pair of vectors in Rn by introducing a weight function with out having any order restriction. We use technique of Lorenz as a tool to introduce stability for a given set of points by assigning a proper weight. We prove the existence of a stable vector for a given set of vectors in Rn with respect to a given weight. Finally, we give examples to illustrate our techniques.

Let be the positive cone of , the set of all vectors of with positive coordinates. Let and be a vector in with Define where We call such a vector as non-vanishing p-vector of Rn.

Let p ∈ Rn and x, y be two non-vanishing p-vectors in Rn. If x ≺p y, then from Figure [1] it is to be noted that Lorenz curve of xp is closer to the Lorenz curve of uniform distribution than the Lorenz curve of yp. In the other words, we say that the vector xp is stable than than the vector yp.

+

Proof. Let p = (p1, p2) and xi = (xi1, xi2) for i ∈ {1, 2, . . . , m}. Then Further, for each there exist a permutation on such that

Consider the set }. As the set has a minimum, let it be at where . Thus we get for all

The following example shows that for n ≥ 3 it is not always possible to find a p-stable vector of a given set of vectors in Rn with positive coordinates and a given vector p in Example 2.4. Take and show that neither majorized by nor majorized by . Thus neither or is a p-stable vector of the set containing the vectors and with respect to the weight p. Example 2.5. Take and Then and . It is easy to verify that neither neither majorized by nor majorized by . Thus neither or is a p- stable vector of the set containing the vectors and with respect to the weight p. Let be a fixed vector in and S = {x1, x2, x3, . . . , xm} be a set if vectors in . Suppose Suppose xi = (xi1, xi2, . . . xin) for i = 1, 2, . . . , m. Then For each there exist a permutation on such that . Consider the set }. This set has a minimum say at where . Thus we get for all

(1) for , where . (2) for , where . (3) for , where .

for , where . If the vectors satisfy these conditions then we get a stable vector say with respect to p.

The authors gratefully thank to the Subhajit Sahaa research scholar of IIT Ropar, for introduction the concept of majoriation and useful discussion comments and constructive suggestions for the betterment of the manuscript.

1. Blackwell, David., Comparison of experiments, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles, 1951. 2. Blackwell, David., Equivalent comparisons of experiments, Annals of Mathematical Statistics, 24, (1953), 93–102. 3. Fuchs, Ladislas.,A new proof of an inequality of Hardy-Littlewood-P´olya, dsskr. B.Mat. Ti, 1947,(1947), 53–54. 4. Peˇcari´c, J. and Abramovich, S., On new majorization theorems, The Rocky Mountain Journal of Mathematics, 27, (1997), 903–911. 5. Ruch, E., Schranner,R. and Seligman, T. H., The mixing distance, The Journal of Chemical Physics 69, (1978), 386-392. 6. Ando, T., Majorization, doubly stochastic matrices, and comparison of eigenvalues,Linear Al- gebra Appl. 118, (1989). 7. Horn, Roger A. and Johnson, Charles R., Matrix analysis, Cambridge University Press, Cam- bridge, (1990).

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