# 臺灣博碩士論文加值系統

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 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.
 圖目錄iii中文摘要ivAbstract v第一章延森不等式11.1 前言. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 延森不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21.2.1 凸函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 等號成立的條件. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 延森不等式的幾何意義. . . . . . . . . . . . . . . . . . . . . . . .41.2.4 凸性確認準則. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.5 支撐線. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 應用. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.1 古典不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.2 代數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.3 三角函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.4 平面幾何. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.3.5 其他不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.4 機率. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.5 競賽題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.6 習題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.6.1 代數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.6.2 三角函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.6.3 平面幾何. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.6.4 其他不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.6.5 機率. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.6.6 競賽題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40第二章舒爾不等式422.1 舒爾不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .422.2 根與係數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .462.3 習題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51第三章Muirhead 不等式533.1 Muirhead 不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . .533.2 兩個有用的技巧. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .633.3 習題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65第四章蓋不等式684.1 舒爾凸函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .684.2 蓋不等式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3 習題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A 簡稱、符號對照表78A.1 簡稱. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.2 符號. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78參考文獻79索引81
 Arnold, B.C. (2007). Majorization: Here, there and everywhere. Statistical Science 22 (3): 407.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 ofthe 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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