一类含迭代的二元均值函数

On a class of two-variable means with iteration

本文给出了含有迭代的二元函数Mf(x,y)=λ1f(x)+λ2f2(x)+μ1f(y)+μ2f2(y)为均值函数的条件,研究了该类均值函数的对称性、等价性及拟算术均值函数关于该类均值函数的不变性,其中f为实区间I上的自映射,f2为f的2次迭代,λ1,λ2,μ1,μ2为实数.

This paper gives the condition that a two-variable function with iteration,i.e.,Mf(x,y)=λ1f(x)+λ2f2(x)+μ1f(y)+μ2f2(y),is mean,wherefmaps a real intervalIinto itself,f2is the second iterate offandλ1,λ2,μ1,μ2are real numbers.Some properties of Mf(x,y)mean type,including the symmetry,equivalence and invariance of the quasi-arithmetic mean with respect to this mean type are discussed.