In the proof of the given 
, a function $\Phi_{3}$ is defined as:$$ \Phi_{3}(x)=\sup \{ \dfrac{\Phi_{2}(xy)}{\Phi_{1}(y)}:y\geq 0\}$$ where $\Phi_{1}$ and $\Phi_{2}$ are $N$- functions with $\Phi_{1}\succ \mid \Phi_{2}$, that is, $\Phi_{1}$ is completely stronger than $\Phi_{2}$.It is written directly that $\Phi_{3}$ is a $N$- function from $\mathbb{R}^{+}$ to $\mathbb{R}^{+}$ . Can somebody tell me how to prove that?.$\Phi_{3}$, by definition, satisfy the following inequality:$$ \Phi_{2}(xy)\leq \Phi_{1}(x)\Phi_{3}(y)~~ \forall x,y\geq 0. ~~ \text{Right?}$$
This Proposition is from the book 'Theory of Orlicz Spaces' by M.M Rao and Z.D.Ren, page $20-21$.