I am reading paper which is mainly about Dobrushin's contraction coefficient and its generalization. In page 27, the following is defined: Consider arbitrary, non-negative, convex function $\psi:\mathbb{R}\mapsto \mathbb{R}$ with $\psi(0)<1$. Then for an integrable random variable $X$ $$||X||_{\psi}:=\inf \{c>0:~\mathbb{E}[\psi(\frac{X}{c})]\leq 1\}.$$
Then letting $\psi^*$ be the convex conjugate of $\psi$, we can write the following for arbitrary random variables $X$ and $Y$ by Young's inequality: $$XY\leq \psi(X)+\psi^*(Y)$$ and hence, $$\mathbb{E}[XY]\leq 2||X||_{\psi}||Y||_{\psi^*}.$$
First of all, in the typical definition of Orlicz norm, it is assumed that $\psi(0)=0$ and also $\psi$ is non-decreasing. Why here those properties are removed?
Secondly, the last inequality is not clear to me. Can anybody give me a hint on how to show that?
Thanks