Let $E\to M$ be a complex vector bundle. A hermitian metric $h$ on $E$ is a hermitian inner product on each fiber $E_{p},\, p\in M$. Suppose that $M$ is also a complex manifold and that $E$ is holomorphic. Under which conditions is $h$ in that case holomorphic? By that I mean that the local matrix representation of $h$ using a local (holomorphic) frame has holomorphic entries. I have never seen commented this condition in the literature, where, if I am not mistaken, one allows for "non-holomorphic" hermitian metrics on holomorphic vector bundles (or at least, that property is not discussed).
Thanks.