suppose we have an Orlicz function $\Phi$ which is an N function and $\Psi$ be its complementary function.
It is given that the following are equivalent
(a) $\Phi \in \Delta_2$ [Which says that $\Phi(2x)\leq K\Phi(x)$ for some $K>0$.]
(b) $\Psi \in \nabla_2$ [Which says that $\Psi(y)\leq \frac{1}{2q} \psi(qx)$ for $q>1.$]If $\Phi \in \Delta_2$ is an N-function then $\Phi(x) \leq C|x|^{\alpha}$ for some $C>0$) and $\alpha>1$ and its complementary function $\Psi \in \nabla_2$ satisfy $\psi(y) \geq D|y|^{\beta}$ for some $D>0$ and $\beta >1$.
Now suppose I assume both $\Phi$ and $\Psi$ are in the intersection of $\Delta_2$ and $\nabla_2$ will the above two properties still hold?
similarly, if suppose there is some property related to the Orlicz space which says that something is true if $\Phi \in \Delta_2$ and suppose I took $\Phi\in \Delta_2 \cap \nabla_2$ will the property still hold?