Let $X$ be a compact Kähler manifold and $(E,h)$ a Hermitian vector bundle over $X$. Suppose that $\nabla$ is a Hermitian-Einstein connection on $E$, that is $$i\Lambda F_\nabla = \lambda\text{id}_E.$$ I'm looking to verify the following formula. If $s$ is a holomorphic section of $E$, then $$\frac{1}{2}\Delta|s|^2=h(i\Lambda F_\nabla(s),s)-|\nabla s|^2.$$
This looks very much like the Bochner formula, but I do not know how to derive this. Is there an analogue of the Bochner formula for compact Kähler manifolds from which coupled with the Hermitian-Einstein condition this follows from?
