On a functional equation arising in the theory of quasiconformal mappings (Q2567990)

The author solves the functional equation \[ {f(x+ h)- f(x)\over f(x)- f(x-h)}= \varphi(h)\psi(x),\quad x\in\mathbb{R},\quad h> 0 \] in the set of injective functions \(f: \mathbb{R}\to \mathbb{R}\), where \(\varphi: (0,\infty)\to \mathbb{R}\) continuous at least one point of \((0,\infty)\), \(\psi: \mathbb{R}\to\mathbb{R}\) continuous and \(\psi(0)= 1\). The solution means a triple \((f,\varphi,\psi)\) which satisfies the considered equation.