It is a classical result that $L^p(\Omega) \subset L^q(\Omega)$ when $q<p$ and $|\Omega| < \infty$. I'd like to know if there is an Orlicz version of this fact. In other words, let $L^{G_1}$ and $L^{G_2}$ be Orlicz spaces. When do we have $L^{G_1} \subset L^{G_2}$?

It seems that this result holds only (maybe) if the Orlicz spaces $L^{G_1} \subset L^{G_2}$ are regular, that is, satisfy the famous $\Delta_2$ and $\nabla_2$ conditions. That is Let's assume the conditions for N-functions introduced by G. Lierbaman (see \cite{Lieberman}). More precisely, $$ G'(t) = g(t), \ \ \text{with} \ \ g \in C^{0}([0,+ \infty]) \cap C^{1}((0,+ \infty]) $$ and for $1< g_{0} \le g_{1} $ fixed constants

If you want to know more about Orlicz Spaces any book about Orlicz Spaces has the concept or articles, for instance see the beginning of Martinez and Wolanski - A minimum problem with free boundary in Orlicz spaces.

More specifically, let $G^p$ and $G^q$ be regular N-functions with $$ 0<g_0^p \le \frac{t(g_p)'(t)}{g_p(t)} \le g_1^p < \infty $$ and $$ 0<g_0^q \le \frac{t(g_q)'(t)}{g_q(t)} \le g_1^q < \infty. $$ Is there a relation between $g_0^p$, $g_1^p$, $g_0^q$, and $g_1^q$ which implies inclusion between $L^{G^p}$ and $L^{G^q}$?

Obs 1 Notice that $p$ and $q$ above are indices not powers.
Obs 2 I asked this in Mathematics StackExchange without answer even when I offered 250 in bounty. I believe now that here is the right place to ask. Any help is welcome, reference, direct proof etc.