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.