  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 $\;Sz(X)=\omega^{\alpha}$, for some
$\alpha<\omega_1$, and $p\in(1,\infty)$, then
  $$Sz(X)\le Dz(X)\le Sz(L_p(X))\le {cases} \omega^{\alpha+1} &\text{if
$\alpha$ is a finite ordinal,}
  \omega^{\alpha} &\text{if $\alpha$ is an infinite ordinal.}
  {cases}$$
