A note on equivalence of means (Q2714377)

Let \(I\) be a real interval, \(M_k : I\times I\to I\) be continuous and internal, that is, \(\min(x,y)<M_k(x,y)<\max(x,y)\) if \(x\neq y\) \((k=1,2)\). Let furthermore \(f: I\to I\) be surjective (``onto''), continuous and satisfy the functional equation \(f(M_1(x,y))=M_2(f(x),f(y))\) \((x,y \in I)\). The authors offer a proof that then \(f\) is also injective (``one-to-one''), thus bijective.