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Jensen Inequality, Muirhead Inequality and Majorization InequalityChen, Bo-Yu 06 July 2010 (has links)
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.
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