Virtanen Ari | Akateeminen Kirjakauppa

This thesis consists of three parts. In the first part, we survey relevant results on majorization and Schur-convexity and produce some auxiliary results needed later. In the second part, we study the set In¡ý of decreasingly ordered n-tuples of elements of a real interval I, under the elementwise (partial) order ¡Ü and the majorization (partial) order preceq. We find the supremums and infimums of the set { x ¡Ê In¡ý | S(x) = a, G(x) = b }, relative to these orders. Here G is a Schur-convex function and S(x) denotes the sum of the elements of x. Besides the constraint G(x) = b, we also consider the constraint G(x) ¡Ü b as well as G(x) ¡Ý b. We tie this discussion to eigenvalue estimation. In the third part, we generalize the majorization order to what we call k-majorization. We find the supremum and infimum relative to ¡Ü of the set {x ¡Ê Rn¡ý | S(x) = a, G(x) = b, x is majorized by c}, where c ¡Ê Rn. We consider the question about the extreme values of the function f(xk, xl) in the {x ¡Ê In¡ý | S(x) = a, G(x) = b}. Particularly, we solve the problem max { xk/xl | x ¡Ê Rn+, ¡Æi xi = a, ¡Çi xi = d }. An equivalent problem is the following: Let A be an n¡Án-matrix with real eigenvalues. Find the best possible upper bound for the ratio of its kth and lth largest positive eigenvalues, using n, tr A, and det A. In this part, we also characterize functions which are increasing relative to 3-majorization. As an application, we find the maximum and minimum of xk subject to x ¡Ê In¡ý, S(x) = a, G(x) = b, and F(x) = c, where F is increasing relative to 3-majorization in the set {x ¡Ê In¡ý | S(x) = a, G(x) = b}. As an example, we present the best possible bounds for the kth largest eigenvalue of A, using besides n, tr A, and tr A2, also either tr A3 or, when the eigenvalues are nonnegative, tr A4.