When a sequence $f \in \ell_1$, there is a very simple bound on its $\ell_q$-norms given by $\|f\|_q^q \leq \|f\|_1 \cdot \|f\|_\infty^{q-1}$.

I am looking for the corresponding inequality for Orlicz norms:

**Question:** Given a Young function $\phi$, and $f \in \ell_1$ is there a bound of the form
 $\|f\|_\phi \leq $ some expression with $\|f\|_1$ and $\|f\|_\infty$ (so that the "expression" tends to 0 if $\|f\|_\infty$ tends to 0 and $\|f\|_1$ remains bounded]).

Naively, one would bound: $\sum \phi(|x|) \leq \sum |x| \frac{\phi(|x|)}{|x|} \leq \big( \sup_x \frac{\phi(|x|)}{|x|} \big) \sum |x|$. But I am quite sure this is neither optimal nor correct (given that the actual norm is defined dually).

**Note** that the inequality I mentioned is a special case of Lyapounov's inequality: let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in \ell_p \cap \ell_q $, then

$$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q}$$ 

So a reference to this larger context could also be nice (how to bound one Orlicz norm from two others)

EDIT: some further comments:

- If it helps remove some technicalities, I would be fine with the extra assumption that $\phi: [0,\infty[ \to [0,\infty[$ is strictly increasing.

- The above inequalities are also true in the $L_p$-spaces (if you add the obvious assumptions like $f \in L_1 \cap L_\infty$). They can be proved using Hölder's inequality (although $\|f\|_q^q \leq \|f\|_1 \|f\|_\infty^{q-1}$ can be proved using a much more naive argument)

- I found Hölder inequalities for Orlicz norms (as well as some inequalities about convolutions), but I could not find the above "interpolation" inequalities.