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EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.

Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$ is concave and increasing and $\int_{B(0,\delta)} \rho^\delta(z)dz =1$.

Is it true that

$f^{-1}\left(\int_{B(0,h)} f\left(\int_{B(0,\delta)} |u(x,z)| \rho^\delta(z)dz\right)\;dx\right)\leq \int_{B(0,\delta)} f^{-1}\left(\int_{B(0,h)} f\left(|u(x,z)| \right)\;dx\right)\rho^\delta(z)dz$?

When $f(t)=t^p$, then $f^{-1}(t)=t^\frac{1}{p}$ and this is precisely Minkowski's Inequality for integrals, as the title suggests, and the proof uses duality in $L^p$. Does this theorem stretch to some class of $N$-functions $f$ ($\Delta_2$, etc.)?

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