齐次可微函数的对角递减性与一类不等式的证明

The diagonal decreasing property of homogeneous differentiable functions and the proof of a class of inequalities

研究了齐次可微函数的对角递减性.对角递减性可以被使用去证明许多不等式,如算术-几何(A-G)平均不等式, Schur不等式, Suranyi不等式等等.文中计算出了对角递减函数在非负三元二次型中出现的概率约为57%.为了弥补对角递减性的不足引入了分块对角递减性的概念.证明了在标准单形上严格正的齐次多项式都是分块对角递减函数.

The diagonal decreasing property of homogeneous differentiable functions(DDF)is investigated in this article.It can be used to prove many inequalities,such as arithmetic geometric(A-G)mean inequality,Schur inequality,and Suranyi inequality.According to the calculation in this paper,the probability of occurrence of diagonal decreasing function in nonnegative ternary quadratic form is about 57%.In order to make up for the deficiency of diagonal decreasing,the concept of block diagonal decreasing is introduced.It is proved that strictly positive homogeneous polynomials on a standard simplex are block diagonal decreasing functions.