商均值的平衡性

On the balancing property of quotient mean

商均值由具有相反严格单调性的两个正值函数衍生而成,是一类更广泛的均值,包括了算术均值、拟算术均值和几何均值.对称性、双对称性及平衡性都是均值的重要性质,对于一般的均值对称和双对称性同时成立则蕴含着平衡性.因此从商均值和平衡性的定义入手,考虑得到了商均值的平衡性与对称性等价.

The quotient mean is derived from two positive functions that have opposite strict monotonicity,and is a generalized mean including arithmetic,quasi-arithmetic and geometric means.Symmetry,bisymmetry and balancing property are all the important properties of means,a mean which is symmetric and bisymmetric must be balanced.From the definitions of balancing property and quotient mean,it get the equivalence of its symmetry and balancing property.