positive upper type $p_{\Phi}^+$. Then there exists a positive constant $C$ such that $$\Phi(t_1+t_2)\le C\lf[\Phi(t_1)+\Phi(t_2)\r],\quad\forall\ t_1,\ t_2\in[0,\infty).$$

Obviously we only need to consider the case when $t_1+t_2>0$. If $p_{\Phi}^+ \in(0,1]$, then, for any $i\in\{1,2\}$, $$\frac{t_i}{t_1+t_2}\Phi(t_1+t_2)\lesssim\Phi(t_i)$$ and hence $$t_i\Phi(t_1+t_2)\lesssim\Phi(t_i)(t_1+t_2),$$ which, via taking the summation on $i$ on both side, further implies the desired conclusion. If $p_{\Phi}^+ \in(1,\infty)$, then let $\widetilde{\Phi}(t):=\Phi(t^{1/p_{\Phi}^+})$ for any $t\in[0,\infty)$. It is easy to check that $\widetilde{\Phi}$ is an Orlicz function of upper type $1$ and hence, by the proved conclusion, we have $$\Phi(t_1+t_2)=\widetilde{\Phi}\lf([t_1+t_2]^{p_{\Phi}^+}\r)\lesssim\widetilde{\Phi}\lf(t_1^{p_{\Phi}^+}\r)
+\widetilde{\Phi}\lf(t_2^{p_{\Phi}^+}\r)
\sim\Phi(t_1)+\Phi(t_2).$$ This finishes the proof of Lemma \[lem1\].

When $\Phi$ is an Orlicz function with positive upper type $p_{\Phi}^+$, from Lemma \[lem1\], it is easy to deduce that $\|\cdot\|_{L^\Phi(\rn)}$ is a quasi-norm.

The following lemma is well known.

\[lem2\] Let $\Phi$ be an Orlicz function with positive lower type $p_{\Phi}^-$ and positive upper type $p_{\Phi}^+$ and $$\widetilde{\Phi}(t):=\int_0^t\frac{\Phi(s)}{s}\,ds,\quad \forall\,t\in(0,\infty).$$ Then $\widetilde{\Phi}$ is also an Orlicz function, which is equivalent to $\Phi$ and $\widetilde{\Phi}$ is continuous and strictly increasing.

\[re\] Observe that all the results stated in this article are invariant under the change of equivalent Orlicz functions. Moreover, equivalent Orlicz functions share the same positive upper and the same lower type numbers. In what 