[0,\infty)\  \rightarrow\  [0,\infty)$ is defined by setting, for any $y\in[0,\infty)$ $$\Psi(y):=\sup\lf\{xy-\Phi(x):\ x\in[0,\infty)\r\}.$$

\[ddd\] A Young function $\Phi:\ [0,\infty)\  \rightarrow\  [0,\infty)$ is called an *N-function* if $\Phi(0)= 0$, $\Phi(t)>0$ for any $t\in(0,\infty)$, $\lim_{t\rightarrow\infty}\frac{\Phi(t)}{t}=\infty$ and $\lim_{t\rightarrow0^+}\frac{\Phi(t)}{t}=0$, here and hereafter, $t\rightarrow0^+$ means $t\in(0,\infty)$ and $t\rightarrow0$.

\[odj\] Let $\Phi$ be an Orlicz function with lower type $p_{\Phi}^-\in(1,\infty)$ and positive upper type $p_{\Phi}^+$. Then there exists an N-function $\widetilde{\Phi}$, which is equivalent to $\Phi$.

Consider the function $$\widetilde{\Phi}(t):=\begin{cases}\displaystyle\int_0^t \sup_{\tau\in(0,s)}\frac{\Phi(\tau)}{\tau}\,ds,\quad\forall\,t \in (0,\infty],\\
\ 0,\quad t=0.
\end{cases}$$ Then it is easy to prove that $\widetilde{\Phi}$ is convex on $[0,\infty)$. By the assumption that $p_{\Phi}^-\in[1,\infty)$, we know that, for any $t\in(0,\infty)$, $$\widetilde{\Phi}(t)\le t\sup_{\tau\in(0,t)}\frac{\Phi(\tau)}{\tau}\le C_{(p_\Phi^-)}t\sup_{\tau\in(0,t)}\lf(\frac{\tau}{t}\r)^{p_\Phi^-}\frac{\Phi(t)}{\tau}\le C_{(p_\Phi^-)}\Phi(t).$$ On the other hand, for any $t\in(0,\infty)$, we have $$\Phi(t)\le C_{(p_\Phi^+)}2^{p_\Phi^+}\Phi(t/2)\le C_{(p_\Phi^+)}2^{p_\Phi^+}\int_{t/2}^{t}\sup_{\tau\in(0,s)}\frac{\Phi(\tau)}{\tau}ds
\le C_{(p_\Phi^+)}2^{p_\Phi^+}\widetilde{\Phi}(t).$$ Thus, we obtain $\Phi\sim\widetilde{\Phi}$. Moreover, it is easy to prove that $\widetilde{\Phi}$ is an N-function, which completes the proof of Lemma \[odj\].

\[nfunction\]

1.  Observe that all the results stated in this article are invariant under the change of equivalent Orlicz functions. In what follows, by Lemma \[odj\] and its proof, without loss of generality, we may *always assume* that an Orlicz function $\Phi$ of lower type $p_\Phi^-\in(1,\infty)$ is also an N-function and an Orlicz function $\Phi$ of lower 