Generalization of the Harmonic Weighted Mean Via Pythagorean Invariance Identity and Application

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 Gg(x,y)=g(x)g(y)y{G_g}\left({x,y} \right) = {{g\left(x \right)} \over {g\left(y \right)}}y generalize, respectively, 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_f,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of H_f,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.