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Let M₁ and M₂ be N-functions. We establish some combinatorial inequalities and show that the product spaces $\ell {ⁿ}_{M₁}\left(\ell {ⁿ}_{M₂}\right)$ are uniformly isomorphic to subspaces of L₁ if M₁ and M₂ are “separated” by a function $t^r$, 1 < r < 2.

Given a normalized Orlicz function M we provide an easy formula for a distribution such that, if X is a random variable distributed accordingly and X₁,...,Xₙ are independent copies of X, then $1/C_p{\left|\right|x\left|\right|}_M\le \left|\right|\left(x_i{X}_i\right){ⁿ}_{i=1}{\left|\right|}_p\le C_p{\left|\right|x\left|\right|}_M$, where $C_p$ is a positive constant depending only on p. In case p = 2 we need the function t ↦ tM’(t) - M(t) to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into L₁[0,1]. We also provide a general result replacing the ${\ell }_p$-norm by an arbitrary N-norm. This...