# Search Results

## Abstract

Under some simple conditions on the real functions *f* and *g* defined on an interval *I* ⊂ (0, *∞*), the two-place functions *A _{f}* (

*x, y*) =

*f*(

*x*) +

*y − f*(

*y*) and

*A*and

*G*, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity

*G*∘ (

*H, A*) =

*G*(equivalent to the Pythagorean harmony proportion), a suitable weighted extension

*H*of the classical harmonic mean

_{f,g}*H*is introduced. An open problem concerning the symmetry of

*H*is proposed. As an application a method of effective solving of some functional equations involving means is presented.

_{f,g}