A question on gauge norm on Orlicz space

Question

I was reading Empirical Processes from "Weak Convergence and Empirical processes" by Van Der Waart and Jon Wellner. There I was studying a norm on a space of random variables defined as follows:

Let $\psi$ be a monotone nondecreasing, convex function with $\psi(0) = 0$ (*). The gauge (Luxemburg) norm of an integrable random variable X wrt $\psi$ is given by

$$\|X\|_{\psi} = \inf \left\{u>0: \mathbb{E}\left[\psi\left(\frac{|X|}{u}\right) \right] \leq 1\right\}$$

As an exercise, the book asks to prove that if $1\leq p\leq q$, then $$\|X\|_{\psi_p} \leq \left(\log 2\right)^{p/q} \|X\|_{\psi_q}$$

where $\psi_p = e^{x^p}-1$ and all logs are natural logs.

My question is how to go about it. I have succeeded in proving the following results which may turn out to be useful here:

a) If $\psi$ and $\phi$ are two functions satisfying the requirement (*), and $\psi \leq \phi$, then $\|X\|_{\psi} \leq \|X\|_{\phi}$.

b) Given a constant $C \geq 1$, $\|X\|_{C\psi} \leq C\|X\|_{\psi} $

c) For $p \geq 1$, $(\log2)^{1/p}\|X\|_{\psi_p}$ is monotone nondecreasing in p. Note: $\psi_p$ as defined above.

d) For $\psi = x^p$, it is the standard $L_p$ norm of X.

I figured it would follow from (c) since the book hints at that but wasn't having much luck with it. I even tried splitting the domain of $e^{x^p}$ into two parts, $x\geq 1$ and $0 \leq x < 1$ and tried manipulating but that got me nowhere.

I appreciate any hints, ideas and even answers on this one.

Please note: One can associate several equivalent norms to an Orlicz space. This post is about the gauge (aka Luxemburg norm), as opposed to the Orlicz norm ( itself a special instance of Amemiya norm), see Hudzik, Henryk; Maligranda, Lech, Amemiya norm equals Orlicz norm in general, Indag. Math., New Ser. 11, No.4, 573-585 (2000). ZBL1010.46031.

Answer 1

In the book, there is a hint, which is the following:

The function $\phi$ for which $\psi_p(x(\log 2)^{1/p})=\phi(\psi_q(x(\log 2)^{1/q}))$ is concave and satisfies $\phi(1)=1$ for $q\geqslant p$. Use Jensen's inequality.

The role of the function $\phi$ is to make a link between what is integrated when we consider $E[\phi_p(|X|/u)]$ and $E[\phi_q(|X|/u)]$ by a function which fulfills the conditions of Jensen's inequality.

Exercice 2 tells us that the constant is optimal taking $X=1$.