  One can associate to a complex projective structure on a surface holomorphic
quadratic differential $\Phi$ via the Schwarzian derivative and a bending
lamination $\lambda$ via the Thurston parameterization. In this note we obtain
upper bounds on the $L^2$-norm of $\Phi$ in terms of the length of $\lambda$.
The proof uses the theory of $W$-volume introduced by Krasnov-Schlenker.
