Schur Power Convexity For Several Types Of Symmetric Functions

The main purpose of this paper is to investigate the Schur power convexity for Lehmer mean of several variables and two classes of symmetric functions.the Schur convexity,Schur geometric convexity and Schur harmonic convexity are exactly the special case of the Schur power convexity,when the m = 1,0,-1.So this paper generalizes the results of Sun[30-31]about the Schur convexity,Schur geometric convexity and Schur harmonic convexity for two classes of symmetric functions.It is well known that there are many classical means of any two positive numbers x and y,for example,which are arithmetic mean A(x,y),geometric mean G(x,y),harmonic mean H(x,y),contra-harmonic mean C(x,y)and so on.However,they are exactly the special case of Lehmer mean of several variables studied in this paper.Therefore,we study the Schur power convexity of Lehmer mean of several variables.In the first chapter,firstly we introduce the research significance and history of the subject,exposing the long history of its development,the wide influence and the key of the role.Then we introduce the definition of Lehmer mean of several variables and two classes of symmetric functions,and the related research result-s of Schur convexity,Schur geometric convexity and Schur harmonic convexity.What'more,the Schur power convexity of some classical functions are introduced.Finally,the conclusion and innovation about Schur power convexity of Lehmer mean of several variables and two classes of symmetric functions are presented.In the second chapter,in preparation for the main research results,and be-cause of two classes of symmetric functions is related to complete symmetric func-tion,we introduce the concept and properties of complete symmetric function in this paper.Furthermore,the lemma about Schur power convexity is presented.In the third chapter,we study the Schur power convexity for Lehmer mean of several variables,which generalizes the results of Gu[26]and Fu[27]about the Schur convexity,Schur geometric convexity and Schur harmonic convexity for Lehmer mean of several variables,and establish some comparison inequalities with power mean of several variables.In the forth chapter,we study the Schur power convexity for Fn(x,r)and Gn(x,r),which generalizes the results of Sun[30-31]about the Schur convexity,Schur geometric convexity and Schur harmonic convexity for Fn(x,r)and Gn(x,r).In the fifth chapter,summarize this paper and expound the problems to be solved in the future.