一类含迭代的二元均值函数

本文给出一类含有迭代的二元函数$M_f(x,y)=\lambda_1f(x)+\lambda_2f^2(x)+\mu_1f(y)+\mu_2f^2(y)$是均值函数的条件，并进一步研究了该类均值函数的对称性、等价性，以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， $f^2$为$f$的$2$次迭代，$\lambda_1,\lambda_2,\mu_1,\mu_2$为实数.

On a class of two-variable means with iteration

This paper gives the condition that a class of two-variable functions with iteration shown in the text are means, i.e. $M_f(x,y)=\lambda_1f(x)+\lambda_2f^2(x)+\mu_1f(y)+\mu_2f^2(y)$, where $f$ maps a real interval $I$ into itself, $f^2$ is the second iterate of $f$, and $\lambda_1,\lambda_2,\mu_1,\mu_2$ are real numbers. In addition, this study discusses the properties of $M_f(x,y)$ mean type, including their symmetry, equivalence, and the invariance of the quasi-arithmetic mean with respect to this mean type.