Cauchy differences and means

Abstract. Basing on the four types of Cauchy differences, some general constructions of
\(k\)-variable premeans and means generated by a single variable real function \(f\) defined in a real interval \(I\) is discussed, special cases are examined and open questions are proposed. In particular, if \(I\) is closed under the addition, and \(f\) is such that \(F\left( x\right) :=f\left(kx\right) -kf\left( x\right)\) is invertible, then the first of four considered functions \(M_{f}:I^{k}\rightarrow \mathbb{R}\) is of the form \begin{equation*} M_{f}\left( x_{1},...,x_{k}\right) =F^{-1}\left( f\left( x_{1}+...+x_{k}\right) -\left( f\left( x_{1}\right) +...+f\left( x_{k}\right) \right) \right) . \end{equation*} Conditions under which \(M_{f}\) is a \(k\)-variable mean (referred to as quasi-Cauchy difference mean of additive type) are examined.