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博碩士論文 etd-0706110-113622 詳細資訊
Title page for etd-0706110-113622
論文名稱
Title
延森不等式、Muirhead 不等式和蓋不等式
Jensen Inequality, Muirhead Inequality and Majorization Inequality
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
92
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2010-06-04
繳交日期
Date of Submission
2010-07-06
關鍵字
Keywords
三弦引理、相異代表系、伯克霍夫定理、凸函數、凹函數、凸包、雙重隨機矩陣、延森不等式、羅倫斯曲線、蓋不等式、蓋、Muirhead 不等式、Muirhead 條件、舒爾凹函數、舒爾凸函數、舒爾準則、舒爾不等式、支撐線不等式、支撐線
system of distinct representatives, Three Chord Lemma, Birkhoff Theorem, convex function, concave function, convex hull, double stochastic matrix, Jensen Inequality, Lorenz Curve, Majorization Inequality, majorization, Muirhead Inequality, Muirhead condition, Schur concave function, Schur Criterion, Schur convex function, Schur Inequality, supporting line, Supporting Line Inequality
統計
Statistics
本論文已被瀏覽 5734 次,被下載 1565
The thesis/dissertation has been browsed 5734 times, has been downloaded 1565 times.
中文摘要
本文第 1 章介紹延森不等式及其幾何意義,並介紹常用的幾個凸性確認準則。同時包含了延森不等式在各領域的應用。
第 2 章的舒爾不等式能簡單的處理三變數對稱不等式的問題。除此之外,也介紹如何利用三次多項式中根與係數的關係將舒爾不等式改寫的方法。
第 3 章由”蓋”的觀念引入Muirhead 不等式,它是算幾不等式的推廣。本章不僅討論蓋與Muirhead 條件的等價關係,同時也介紹兩個應用上的小技巧。
第 4 章討論蓋不等式,它與不等式理論中最多產的蓋\及舒爾凸函數相關,並考慮在基本對稱函數、樣本變異數、熵及生日問題上的應用。
Abstract
Chapter 1 introduces Jensen Inequality and its geometric interpretation. Some useful criteria for checking the convexity of functions are discussed. Many applications in various fields are also included.
Chapter 2 deals with Schur Inequality, which can easily solve some problems involved symmetric inequality in three variables. The relationship between Schur Inequality and the roots and the coefficients of a cubic equation is also investigated.
Chapter 3 presents Muirhead Inequality which is derived from the concept of majorization. It generalizes the inequality of arithmetic and geometric means.
The equivalence of majorization and Muirhead’s condition is illustrated. Two useful tricks for applying Muirhead Inequality are provided.
Chapter 4 handles Majorization Inequality which involves Majorization and Schur convexity, two of the most productive concepts in the theory of inequalities.
Its applications in elementary symmetric functions, sample variance, entropy and birthday problem are considered.
目次 Table of Contents
圖目錄iii
中文摘要iv
Abstract v
第一章延森不等式1
1.1 前言. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 延森不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
1.2.1 凸函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 等號成立的條件. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 延森不等式的幾何意義. . . . . . . . . . . . . . . . . . . . . . . .4
1.2.4 凸性確認準則. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.5 支撐線. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 應用. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 古典不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 代數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 三角函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.4 平面幾何. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.5 其他不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4 機率. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5 競賽題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.6 習題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.6.1 代數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.6.2 三角函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.6.3 平面幾何. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.6.4 其他不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.6.5 機率. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.6.6 競賽題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
第二章舒爾不等式42
2.1 舒爾不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42
2.2 根與係數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46
2.3 習題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
第三章Muirhead 不等式53
3.1 Muirhead 不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
3.2 兩個有用的技巧. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
3.3 習題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
第四章蓋不等式68
4.1 舒爾凸函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68
4.2 蓋不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 習題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A 簡稱、符號對照表78
A.1 簡稱. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
A.2 符號. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
參考文獻79
索引81
參考文獻 References
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Clevenson, M.L. and Watkins, W. (1991). Majorization and the birthday inequality. Mathematics Magazine 64 (3): 183-188.
Cosnita, C. and Fanica, T. (1966). Culegere de probleme de matematici pentru examenele de maturitate si admitere in invatamintul superior. Bucuresti: Editura Tehnica.
Daykin, D.E. (1969). Problem 5685. Amer. Math Monthly 76, 835; see also Amer. Math. Monthly 77 (1970), 782.
Joag-Dev, K. and Proschan, F. (1992). Birthday problem with unlike probabilities. Amer. Math. Monthly 99 (1): 10-12.
Klamkin, M.S. (1970). A physical application of a rearrangement inequality. Amer. Math. Monthly 77 (1): 68-69.
Klamkin, M.S. (1975). Extensions of the weierstrass product inequalities. II. Amer. Math. Monthly 82, 741-742.
Lorenz, M.O. (1905). Methods of measuring the concentration of wealth. Publications of
the American Statistical Association 9 (70): 209-219.
Marshall, A.W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. New York: Academic Press.
Muirhead, R.F. (1903). Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proc. Edinburgh Math. Soc 21, 144-157.
Nasser, J.I. (1966). Problem E847. Amer. Math. Monthly 73, 82; see also Amer. Math. Monthly 77 (1970) 524.
Ross, S.M. (2008). A First Course in Probability, 8th Edition. Harlow: Pearson Education.
Schur, I. (1923). Uber eine Klasse von Mittelbildungen mit Anwendungen die Determinanten-Theorie Sitzungsber. Berlin. Math. Gesellschaft 22, 9-20 Issai Schur Collected Works (A. Brauer and H. Rohrbach, eds.) Vol. II. PP. 416-427. Berlin: Springer-
Verlag, 1973.
Steele, J.M. (2004). The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge: Cambridge University Press.
黃宣國(1991),凸函數與琴生不等式,上海: 上海教育出版社。
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