On a functional equation arising in the theory of quasiconformal mappings

Abstract

We solve functional equations of the form $$k(x,h) = \frac{{f(x + h) - f(x)}}{{f(x) - f(x - h)}},\quad x \in {\mathbb{R}},\;h > 0,$$ in the class of injective functions \(f:{\mathbb{R}}: \to {\mathbb{R}},\) provided \(k:{\mathbb{R}} \times (0,\infty ) \to {\mathbb{R}}\) is a product of two functions, each of them of one variable.