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In the Handbook of the Geometry of Banach Spaces, vol 1, page 855, Johnson and Schechtman say that if $M$ is an Orlicz function, the following are equivalent:

  • The unit vector basis of the Orlicz space $\ell_M$ is $p$-convex and $2$-concave;
  • $M(\left|t\right|^\frac{1}{p})$ is equivalent to a convex function and $M(t^{\frac{1}{2}})$ is equivalent to a concave function on $[0, \infty)$

And they give as reference Application de l'étude de certaines formes linéaires aléatoires au plongement d'espaces de Banach dans des espaces $L^p$, by J. Bretagnolle and D. Dacunha-Castelle, which I cannot understand since I don't speak French.

So my question is: how does one prove this result?

Thank you!

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  • $\begingroup$ Would you mind including, what an Orlicz-function is? (I could think of different meanings…) $\endgroup$ – Dirk Nov 10 '17 at 11:22
  • $\begingroup$ An Orlicz function $M : \mathbb{R} \rightarrow \mathbb{R}$ is an even convex function such that $M(0) = 0$ and $\lim_{t \rightarrow \infty} M(t) = \infty$. $\endgroup$ – Seven9 Nov 10 '17 at 12:11
  • $\begingroup$ Sorry, I forgot to tell that $M(t) \geq 0$ for every $t$. $\endgroup$ – Seven9 Nov 10 '17 at 23:57

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