We consider the invariance of the arithmetic mean with respect to some Cauchy mean-type mapping, namely, we present some results concerning the functional equation

$$\begin{aligned} \left( \frac{f'}{g'}\right) ^{-1}\left( \frac{f(x)-f(y)}{g(x)-g(y)}\right) +\left( \frac{h'}{k'}\right) ^{-1}\left( \frac{h(x)-h(y)}{k(x)-k(y)}\right) =x+y, \quad x\ne y, \ x,y\in I, \end{aligned}$$

where \(I\subset (0,+\,\infty )\) is an open interval, \(g,k:I\rightarrow {\mathbb {R}}\) are power functions, \(f,h:I\rightarrow {\mathbb {R}}\) are differentiable functions such that \(\frac{f'}{g'}\) and \(\frac{h'}{k'}\) are injective functions.

中文翻译：

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一些算术平均值不变的柯西均值类型映射

我们考虑算术平均值关于某些柯西均值类型映射的不变性，即，我们给出有关函数方程的一些结果

$$ \ begin {aligned} \ left（\ frac {f'} {g'} \ right）^ {-1} \ left（\ frac {f（x）-f（y）} {g（x）- g（y）} \ right）+ \ left（\ frac {h'} {k'} \ right）^ {-1} \ left（\ frac {h（x）-h（y）} {k（x ）-k（y）} \ right）= x + y，\ quad x \ ne y，\ x，y \ in I，\ end {aligned} $$

其中\（I \ subset（0，+ \，\ infty）\）是一个开放区间，\（g，k：I \ rightarrow {\ mathbb {R}} \）是幂函数，\（f，h： I \ rightarrow {\ mathbb {R}} \）是微分函数，因此\（\ frac {f'} {g'} \）和\（\ frac {h'} {k'} \）是内射函数。