Some Cauchy mean-type mappings for which the arithmetic mean is invariant

Abstract

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.