Around a problem of Nicole Brillouët-Belluot (Q744042)

The article deals with the functional equations \[ f(x) \cdot f^{-1}(x) = x^\alpha \qquad (x \in I), \] where \(I \subset {\mathbb R}\) is a nontrivial interval, \(\alpha \in {\mathbb R}\), \(f:\;I \to I\) is a bijection. In the case \(I = (0,\infty)\), \(\alpha \geq 2\), this equation has a trivial solution \[ f(x) = \begin{cases} x^\beta & \text{if} \;0 < x < 1, \\ x^\gamma & \text{if} \;1 < x < \infty, \end{cases} \] where \(\beta\) and \(\gamma\) satisfy the equation \[ \beta + \frac1\beta = \gamma + \frac1\gamma = \alpha. \] The main result is the following: the equation under consideration has no other continuous and increasing solutions. A similar result is proved for decreasing solutions; some cases, when \(I \neq (0,\infty)\), are also considered.