Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$

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 $$\lVert f \rVert_{L^{\Phi}} := \inf \left\{\gamma\,\middle|\,\gamma>0,\,\int_{\Omega} \Phi\left(\frac {\lvert f(x)\rvert}{\gamma} \right)\,\mathrm{d}x\leq 1\right\}.$$

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!

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 I was unable to find an appropriate result.
• I have asked this question on Math.Stackexchange without luck, so I'm trying here.
• Meanwhile he/she changed the name. I have deleted my comment. Jun 13 '19 at 12:36
• @JochenWengenroth: Thanks for being honest with me! Jun 13 '19 at 12:38

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 $$t$$.