The Optimal Control Inequalities And The Schur Convexity Of Some Symmetric Functions

In this paper we mainly do some research about two problems: firstly, we dis-cuss two classes of conjectures of abstract control inequalities which were proposedby Dinghua Yang in/Theory Foundation of Abstract Control Inequalities0,andthen we proof that the first conjecture is efective under the condition of theweighted arithmetic mean. Similarly,we also proof that the second conjecture isefective under the condition of the arithmetic mean. Secondly, we research theschur convexity and schur harmonic convexity of two classes of symmetric func-tions, establish some results. As applications, we obtain some new inequalities.The structure of this paper is as follow.In chapter1,we briefly introduce the relative background and the presentresearch situation of abstract control inequalities,schur convexity functions andschur harmonic convexity functions.In addition, we introduce the main work ofthis article.In chapter2,we introduce some definitions and natures about control in-equality, abstract average, symmetric function, schur convexity, schur harmonicconvexity. Besides,we introduce some important lemmas.In chapter3,under the condition of the weighted arithmetic mean or arith-metic mean,we choose suitable abstract Σ(?)Σ’(Σ|-(?)Σ’) down convexity func-tions and use the nature of abstract average to proof the two classes of conjectureswhich were proposed by Dinghua Yang in/Theory Foundation of Abstract Con-trol Inequalities0are correct.In chapter4,we research the schur convexity and schur harmonic convexityof two classes of symmetric functions and establish a few main results,what’smore,we give some relative applications.