Norm of Laplacian of a differential form

Let $M$ be a closed Riemannian manifold and $\beta$ be a $p$-form on $M$. So, the Hodge-Laplacian $\Delta\beta$ is again a $p$-form. I want to know if we have any relation or inequality regarding $\Delta|\beta|^2$ and $\Delta\beta$.

Proposition (Bochner-weitzenbock formula). For any $p$-form $\beta$, $$\frac{1}{2}\Delta|\beta|^2=|\nabla \beta|^2-\langle\Delta \beta,\beta\rangle+F(\beta),$$ where $F$ is a complicate expression related to curvature tensor.

For proof and more information see

Poor, Walter A., Differential geometric structures, New York etc.: McGraw-Hill Book Company. XIII, 338 p. DM 102.13 (1981). ZBL0493.53027.

and

Petersen, Peter, Riemannian geometry, Graduate Texts in Mathematics. 171. New York, NY: Springer. xvi, 432 p. (1998). ZBL0914.53001.