Theory Of Majorization And Generalized Power Convex Functions

Majorization permeates almost every branch of mathematics and plays a wonderful role everywhere.In research of the theory of majorization,there are two important and fundamental objects:establishing majorizing relations among vectors and finding various Schur-convex functions.Majorizing relations deeply characterize intrinsic connections among vectors and combining a majorizing relation with suitable Schur-convex funcitons can lead to various interesting inequalities.The convex function is inseparable from the Schur-convex function.A well-known conclusion is that the multivariate symmetric convex function is a Schur-convex func-tion.In recent years,the study of generalized convex functions is a very active topic.Combining some generalized convex functions and the Schur-convex function theory can construct some new Schur-convex function theories,such as geometrically convex func-tion and Schur-geometrically convex function theory,harmonically convex function and Schur-harmonically convex function theory.The main results of this dissertation are as follows:Firstly,we prove a majorization inequality involving the cyclic moving average,which is a majorizing relation between two vectors generated by a monotonically decreas-ing vector.Then an open problem,which was proposed by Professor Ingram Olkin in 2006,is solved.Secondly,we introduce the arithmetic m-power convex function.Ifm = 1,then the arithmetic m-power convex function is convex function.And we prove that the multivari-ate symmetrical arithmetic m-power convex function is Schur-convex function.Moreover,some properties of the arithmetic m-power convex function are studied.Finally the Schur-convexity for the elementary symmetric composite functions and its inverse problem are discussed,and several inequalities involving arithmetic means are established.Then,we introduce the geometric m-power convex function and harmonic m-power convex function.If m = 0,then the geometric m-power convex function is geometrically convex function.If m =-1,then the harmonic m-power convex function is harmonically convex function.We prove that the multivariate symmetrical geometric(harmonic)m-power convex function is Schur-geometrically(harmonically)convex function.Moreover,some properties of the geometric(harmonic)m-power convex function are studied.Finally the Schur-geometric(harmonic)convexity for the elementary symmetric composite func-tions and its inverse problem are discussed,and several inequalities involving geometric and harmonic means are established.Lastly,we introduce the conception of generalized power convex function,and we denote Mm1Mm2-convex function.When mi = 1,0,-1,respectively,Mm1Mm2-convex function correspond to arithmetic m2-power convex function,geometric m2-convex func-tion and harmonic m2-convex function.And we prove that the multivariate symmetrical Mm1m2-convex function is Schur m2-power convex function.Moreover,some properties of the arithmetic Mm1Mm2-convex function are studied.