关于广义Schur-凸函数及其应用的研究

The theory of majorization inequalities is mainly to study the majorized relations of the sum of components of a real vector, the Schur-convexity of real and multivariable functions, and their applications. Schur-convex functions are generalizations of Jensen convex functions and monotonic functions. Majorization inequalities have extensive applications in geometry, graph theory, probability, statistics, optimization, and the like. In recent years, many new concepts and properties concerning Schur-convexity have been being continuously published in mathematical journals in the world and these results have been being extensively applied. In this project, we will define two new concepts of generalized Schur-convex functions, introduce a new mean, and study their various Schur-convexity. As applications, we will employ the newly defined mean to refine inequalities due to Ky Fan and Wang-Wang. Moreover, with the help of homogeneous polynomials, we will investigate various convexity of many multivariable functions, establish inequalities for many newly defined convex functions, and find some analytic inequalities and some matrix spectral inequalities. Our research methods, approaches, and techniques are analytic and constructive. These results obtained in this project will enrich the theoretic system in the field and make the coming study more active and more deep. By this, we will foster the independent research ability of young teachers and graduates, and finally promote academic quantity and quality in Inner Mongolia.

控制不等式理论主要研究实向量分量之和的控制关系、多元实函数的Schur凸性及其应用问题。Schur凸函数是Jensen型凸函数和增函数的推广。控制不等式在几何学、图论、概率数理及最优化等学科中有着广泛的应用。近些年来，国内外学术期刊上不断出现新型的Schur-凸性概念和相关问题的研究成果，且这些成果得到了广泛的应用。 ..在本项目中，我们将定义两个新的广义Schur凸性的概念，引入一种新的平均数，并研究它的各种Schur凸性。作为应用，我们将借助新引入的平均数加细Ky Fan不等式和王-王不等式。另外，我们还要用齐次多项式来研究若干多元函数的各种Schur凸性，建立若干新凸函数的积分不等式，给出一些解析不等式和矩阵谱不等式。我们将用分析和构造性方法研究这些问题。这些成果的获得将丰富该理论体系，使我们的研究工作更加深入，并培养青年教师和研究生的科研能力，推动内蒙地区科研水平的提高。

控制不等式理论主要研究实向量分量之和的控制关系、多元实函数的Schur-凸性及其应用问题。控制不等式在其它学科中有着广泛的应用。近些年来，国内外学术期刊上不断出现新型的Schur-凸性概念和相关问题的研究成果，且这些成果得到了广泛的应用。. 在本项目研究工作引进了两类Schur-凸性概念：几何-算术-Schur-凸函数概念和几何-调和-Schur-凸函数，并给出了几个简单平均数的几何-算术-Schur-凸性和几何-调和-Schur-凸性问题；(2) 建立了一种新的平均数，并研究新平均数的Schur-凸性问题，并用新平均数加细了Ky Fan型不等式和王-王型不等式问题，同时研究了若干个简单平均数的差、商的m-Schur-凸性问题和齐次多项式的m-Schur-凸性问题； (3) 定义了新型广义凸函数概念：(α,m)-几何凸函数、 (α,m)-对数凸函数、s-对数凸函数、广义s-凸函数、几何r-凸函数、算术调和凸函数以及多元函数的分量具有不同凸性的广义协同凸函数概念，并研究相应的积分估计问题。(4) 课题组还研究了复矩阵的对角占优的H-矩阵问题，先后引进了严格几何ε-对角占优矩阵概念及其判断的充分必要条件和ε-局部双对角占优矩阵概念及其判定条件等以及Hermitian矩阵乘积的特征值不等式问题。. 项目组所得成果将丰富该理论体系，并为我们科研团队的将来学术研究工作打下了良好的基础。