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THE CAUCHY – SCHWARZ MASTER CLASS - PART 13

Majorization and Schur Convexity Majorization và lồi Schur là hai trong số các khái niệm hiệu quả nhất trong lý thuyết của sự bất bình đẳng. Họ thống nhất sự hiểu biết của chúng ta giới hạn quen thuộc, và họ chỉ cho chúng tôi vào bộ sưu tập lớn của kết quả chỉ lờ mờ cảm nhận mà không cần sự giúp đỡ của họ. Mặc dù majorization và lồi Schur một vài đoạn văn để giải thích, người ta tìm thấy với kinh nghiệm rằng cả hai khái niệm tuyệt đơn giản | 13 Majorization and Schur Convexity Majorization and Schur convexity are two of the most productive concepts in the theory of inequalities. They unify our understanding of many familiar bounds and they point us to great collections of results which are only dimly sensed without their help. Although ma jorization and Schur convexity take a few paragraphs to explain one finds with experience that both notions are stunningly simple. Still they are not as well known as they should be and they can become one s secret weapon. Two Bare-Bones Definitions Given an n-tuple 7 71 72 7n we let Y j 1 j n denote the jth largest of the n coordinates so 7 1 max 7j 1 j n and in general one has 7 1 7 2 7 n Now for any pair of real n-tuples a a1 a2 an and Ị3 p1 32 Ị3n we say that a is majorized by Ị3 and we write a Ị3 provided that a and Ị3 satisfy the following system of n 1 inequalities a 1 3 1 a 1 a 2 Ạ1 fi 2 . . . a 1 a 2 a n-1 Ạ1 p 2 P n-1 together with one final equality a 1 a 2 a n Ạ1 p 2 p n Thus for example we have the majorizations 1 1 1 1 2 1 1 0 3 1 0 0 4 0 0 0 13.1 and since the definition of the relation a Ị3 depends only on the 191 192 Majorization and Schur Convexity corresponding ordered values a j and P j we could just as well write the chain 13.1 as 1 1 1 1 Y 0 1 1 2 Y 1 3 0 0 Y 0 0 4 0 . To give a more generic example one should also note that for any ai a2 . an we have the two relations a a . . a -Y ai a2 . . On Y ai a2 an 0 . . 0 where as usual we have set Õ a1 a2 . an n. Moreover it is immediate from the definition of majorization that relation -Y is transitive a -Y p and p -Y Y imply that a -Y Y. Consequently the 4-chain 13.1 actually entails six valid relations. Now if Ac Rd and f A R we say that f is Schur convex on A provided that we have f a f p for all a p cA for which a Y p. 13.2 Such a function might more aptly be called Schur monotone rather than Schur convex but the term Schur convex is now firmly rooted in tradition. By the same custom if the first .