Means of Cauchy’s difference type

Abstract

k-variable means which are the Cauchy differences of additive type generated by a real single variable function f, and denoted by \(C_{f,k}\), are examined. It is shown that \(C_{f,k}\) is an increasing mean in \(\left( 0,\infty \right) \) iff f is a convex solution of the (reflexivity) functional equation \(f\left( kx\right) -kf\left( x\right) =x\), and a construction of a large class of such means is presented. The form of a unique homogeneous mean of the form \(C_{f,k}\) is given. As corollaries, the suitable results for the Cauchy differences of exponential, logarithmic and multiplicative types are obtained. It is shown that there exists a unique continuous and differentiable at 0 function f such that \(M\left( x,y\right) :=f\left( x+y\right) -f\left( x\right) f\left( y\right) \) is a bivariable premean in \(\mathbb {R}\), and its analyticity is proved. Finding the explicit form of f is one of the proposed open questions.