# Lyapounov's inequality for Orlicz norms

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

This inequality is a special (or rather limit) 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

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

I am looking for the corresponding inequality for Orlicz norms:

Question: Given Young functions $$\phi_1, \phi_2$$ and $$\phi_3$$, and $$f \in \cap_i \ell_{\phi_i}$$ is there a bound of the form $$\|f\|_{\phi_2} \leq$$ some expression with $$\|f\|_{\phi_1}$$ and $$\|f\|_{\phi_3}$$ (so that the "expression" tends to 0 if one of the norms tends to 0 and while the other remains bounded).

The limit case (where $$\phi_1$$ and $$\phi_3$$ are the 1 and $$\infty$$ norms) could be done as follows (under the assumption that $$\frac{\phi(x)}{x}$$ is increasing and does not take the value $$+\infty$$). It's easier to do it with the Luxemburg norm (which is the same as the original one up to a factor of 2). Let $$f \in \ell_1 \cap \ell_\infty$$, then $$\| f\|_\phi = \inf \lbrace b >0 \mid \sum_n \phi( \tfrac{|f(n)|}{b}) \leq 1 \rbrace$$ Bound: $$\sum \phi(|x|) \leq \sum |x| \tfrac{\phi(|x|)}{|x|} \leq \big( \sup_x \frac{\phi(|x|)}{|x|} \big) \sum |x|$$. Assuming that $$\phi(x)/x$$ is increasing, this means that

$$\sum_n \phi( \tfrac{|f(n)|}{b}) \leq \frac{\phi(\|f\|_\infty / b)}{\|f\|_\infty} \|f\|_1.$$

In particular, if $$b = \dfrac{\|f\|_\infty}{ \phi^{-1} \big(\tfrac{\|f\|_\infty}{\|f\|_1} \big) }$$, where $$\phi^{-1}$$ is a inverse (or semi-inverse if $$\phi$$ is not strictly increasing) of $$\phi$$, the sum is smaller than one. Hence $$\|f\|_\phi \leq \dfrac{\|f\|_\infty}{ \phi^{-1} \big(\tfrac{\|f\|_\infty}{\|f\|_1} \big) }$$

• If it helps remove some technicalities, I would be fine various extra assumptions such as $$\phi_i: [0,\infty[ \to [0,\infty[$$ are 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 [similar to the one above])