Let $\Phi$ be a *Youngs's function*, i.e.
$$ \Phi(t) = \int\_0^t \varphi(s) \,\mathrm d s$$
for some $\varphi$ satifying


1. $\varphi:[0,\infty)\to[0,\infty]$ is increasing
2. $\varphi$ is lower semi continuous
3. $\varphi(0) = 0$
4. $\varphi$ is neither identically zero nor identically infinite


and define the *Luxemburg norm* of $f:\Omega\to\mathbb{R}$ as
$$ \lVert f \rVert\_{L^{\Phi}} := \inf \left\{\gamma\,\middle|\,\gamma>0,\,\int\_{\Omega} \Phi\left(\frac {\lvert f(x)\rvert}{\gamma} \right)\,\mathrm{d}x\leq 1\right\}.$$




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> 
> **Question**: What can we say about $\Phi\left(\lVert f \rVert\_{L^{\Phi}}\right)$? In particular, I'd like to know, if
>  $$\Phi\left(\lVert f \rVert\_{L^{\Phi}}\right) \leq C \int\_{\Omega}\Phi(\lvert f(x)\rvert) \,\mathrm d x$$
>  holds for some $C$
>  independent of $f$.
> 
> 
> 


Any idea or hint for a reference is welcome!




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**Notes**:


* The above inequality trivially holds for $\Phi(t) = t^p$, where $p>1$
* Maybe it's appropriate to consider this question in the more general framework of Musielak-Orlicz spaces. However, e.g. in [Lebesgue and Sobolev Spaces with Variable Exponents](http://www.mathematik.uni-muenchen.de/~diening/archive/book.pdf) I was unable to find an appropriate result.
* I have asked this question on [Math.Stackexchange](https://math.stackexchange.com/questions/3251777/luxemburg-norm-as-argument-of-youngs-function-phi-left-lvert-f-rvert-l) without luck, so I'm trying here.



Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$
The conjectured inequality does not hold.


For a counterexample, consider $\Phi(t)=\max(t^2,t^3)$ and $\Omega=(0,1)$.
Let $f=a\chi\_{(0,b)}$ for $a,b\in (0,1)$.
It can be calculated that $\|f\|\_{L^\Phi}= a b^{1/3}$.
Then the inequality can be written as
\begin{equation\*}
 a^2 b^{2/3} \leq C a^2 b
\end{equation\*}
which is not possible for a constant $C$ independent of $b$.


Intuitively, this is because the 
left-hand side is mostly determined by the values of $\Phi$ for large $t$,
whereas the right-hand side is (for small $a,b$) independent of the values of $\Phi$
for large $t$.


