Mean value property and associated functional equations (Q1306325)

The author offers as main result the solution of the functional equation \[ \frac{g(x)-g(y)}{x-y}=\phi(\frac{f(x)-f(y)}{x-y}). \] If \(f\) is strictly convex (from above or below) then \(\phi\) is affine and \(g(t)\) is a linear combination of \(f(t),t\) and \(1.\) The result is then applied to determine \(g\) and \(h\) in \([g(x)-g(y)]/(x-y)=h[M(x,y)]\) for \(x\neq y\) and for different mean values \(M,\) in particular for the means studied by \textit{K. B. Stolarsky} [Math. Mag. 48, 87-92 (1975; Zbl 0302.26003)].