The Szlenk Index of L_p(X)

We find an optimal upper bound on the values of the weak$^*$-dentability index $Dz(X)$ in terms of the Szlenk index $Sz(X)$ of a Banach space $X$ with separable dual. Namely, if $\;Sz(X)=ω^α$, for some $α<ω_1$, and $p\in(1,\infty)$, then $$Sz(X)\le Dz(X)\le Sz(L_p(X))\le {cases} ω^{α+1} &\text{if $α$ is a finite ordinal,} ω^α &\text{if $α$ is an infinite ordinal.} {cases}$$

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