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. Lieberman (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 $$ 0 < g_0 \le \frac{t g'(t)}{g(t)} \le g_1,\quad\forall t > 0. $$

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.

Is this correct?

\begin{eqnarray} \frac{tg'(t)}{g(t)} \le g_1 \Rightarrow \frac{g'(t)}{g(t)} \le \frac{g_1}{t} \end{eqnarray} As $g:R^+ \longrightarrow R^+$ saisfy $g(t)>0$ and is a $C^{1}(0,\infty) $ nondecreasing, we have $g'(t) \ge 0$ and integrating, we obtain \begin{eqnarray} ln(g(t)) \le ln( t^{g_1}) \Rightarrow g(t) \le t^{g_1} \end{eqnarray} Analogouslly, we get $ g^q(t) \le t^{g_1^q}$ and $t^{g_0^{p}} \le g^p(t) $. Observing that

1. $t^{g_1^q} \le t^{g_0^{p}}$ for $t>1 $ if $g_1^q \le g_0^{p}$. In this case $ g^q(t) \le g^p(t) $
2. Orlicz class are equal Orlicz Spaces if and only if the $N$-function $G \in \Delta_2$
3. The Orlicz classes $L^{G^p}$ and $L^{G^q}$ given respectivally by the N-functions $G^p$ and $G^q$ satisfy $L^{G^p} \subset L^{G^q} $ if and only if there is constantes $a,b$ so that $$ G^q(t) \le aG^p(t) \quad \mbox{for all} \quad t\ge b. $$ we conclude that $L^{G^p} \subset L^{G^q} $ if $g_1^q \le g_0^{p}$

Obs1 I do not know if does hold the reciproc (this is less important for me) Obs2 I do not know much about Orlicz Spaces, so it is possíble that it is a classical result that can be found or deduced in books. Because of this I need a proof verification