A bound on the $L^2$-norm of a projective structure by the length of the bending lamination

One can associate to a complex projective structure on a surface holomorphic quadratic differential $\mathrm{\Phi }$ 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 $\mathrm{\Phi }$ in terms of the length of $\lambda$. The proof uses the theory of $W$-volume introduced by Krasnov-Schlenker.