Subcommutativity of integrals and quasi-arithmetic means

Let $(X,\mathcal{L},\lambda )$ and $(Y,\mathcal{M},\mu )$ be finite measure spaces for which there exist $A\in \mathcal{L}$ and $B\in \mathcal{M}$ 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 $f,g\text{\hspace{0.17em}⁣}:I\to {\mathbb{R}}_{+}$ are homeo-morphisms with $g$ increasing. We prove that the functional inequality $$f^{-1}\text{\hspace{0.17em}⁣}\left({\int }_{X}f\text{\hspace{0.17em}⁣}\left(g^{-1}\text{\hspace{0.17em}⁣}({\int }_{Y}g\circ hd\mu )\right)d\lambda \right)\text{\hspace{0.17em}⁣}\le g^{-1}\text{\hspace{0.17em}⁣}\left({\int }_{Y}g\text{\hspace{0.17em}⁣}\left(f^{-1}\text{\hspace{0.17em}⁣}({\int }_{X}f\circ hd\lambda )\right)d\mu \right)$$ is satisfied by every $\mathcal{L}\otimes \mathcal{M}$-measurable simple function $h:X\times Y\to I$ 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.