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'm not quite sure this is optimal (or even 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)