The Szlenk Index of L_p(X)

We find an optimal upper bound on the values of the weak${}^{\ast }$-dentability index $Dz(X)$ in terms of the Szlenk index $Sz(X)$ of a Banach space $X$ with separable dual. Namely, if $Sz(X)={\omega }^{\alpha }$, for some $\alpha <{\omega }_1$, and $p\in (1,\mathrm{\infty })$, then Sz(X)\le Dz(X)\le Sz(L_p(X))\le {cases} \omega^{\alpha+1} &\text{if $\alpha$ is a finite ordinal,} \omega^{\alpha} &\text{if $\alpha$ is an infinite ordinal.} {cases}