English

On a functional equation related to two-variable Cauchy means

Classical Analysis and ODEs 2020-11-23 v1

Abstract

In this paper, we are dealing with the solution of the functional equation φ(x+y2)(f(x)f(y))=F(x)F(y), \varphi\Big(\frac{x+y}2\Big)(f(x)-f(y))=F(x)-F(y), concerning the unknown functions φ,f\varphi,f and FF defined on a same open subinterval of the reals. Improving the previous results related to this topic, we describe the solution triplets (φ,f,F)(\varphi,f,F) assuming only the continuity of φ\varphi. As an application, under natural conditions, we also solve the equality problem of two-variable Cauchy means and two-variable quasi-arithmetic means.

Cite

@article{arxiv.1807.00662,
  title  = {On a functional equation related to two-variable Cauchy means},
  author = {Tibor Kiss and Zsolt Páles},
  journal= {arXiv preprint arXiv:1807.00662},
  year   = {2020}
}
R2 v1 2026-06-23T02:48:09.893Z