Let \((X, \mathscr {L}, \lambda )\) and \((Y, \mathscr {M}, \mu )\) be finite measure spaces for which there exist \(A \in \mathscr {L}\) and \(B \in \mathscr {M}\) with either \(0< \lambda (A)< 1 < \lambda (X)\) and \(0< \mu (B) < \mu (Y)\) , or the other way around. In addition, let \(I \subseteq \mathbb {R}\) be a non-empty open interval, and suppose that \(f,g:I \rightarrow \mathbb {R}_{+}\) are homeomorphisms with g increasing. We prove that the functional inequality \( f^{-1}\!\left( \int _X f\!\left( g^{-1}\!\left( \int _Y g \circ h\;d\mu \right) \right) d \lambda \right) \! \le g^{-1}\!\left( \int _Y g\!\left( f^{-1}\!\left( \int _X f \circ h\;d\lambda \right) \right) d \mu \right) \) is satisfied by every \(\mathscr {L} \otimes \mathscr {M}\) -measurable simple function \(h: X \times Y \rightarrow I\) if and only if \(f=a \,g^b\) for some \(a,b \in \mathbb {R}_{+}\) with \(b\ge 1\) . An analogous characterization is given for probability spaces.
