  Let $X$ be a complex manifold, $(E,h)\to X$ be a rank $r$ holomorphic
hermitian vector bundle, and $\rho$ be a sequence of dimensions $0 = \rho_0 <
\rho_1 < \cdots < \rho_m = r$. Let $Q_{\rho,j}$, $j=1,\dots,m$, be the
tautological line bundles over the (possibly incomplete) flag bundle
$\mathbb{F}_{\rho}(E) \to X$ associated to $\rho$, endowed with the natural
metrics induced by that of $E$, with Chern curvatures $\Xi_{\rho,j}$. We show
that the universal Gysin formula \textsl{\`{a} la} Darondeau--Pragacz for the
push-forward of a homogeneous polynomial in the Chern classes of the
$Q_{\rho,j}$'s also hold pointwise at the level of the Chern forms
$\Xi_{\rho,j}$ in this hermitianized situation. As an application, we show the
positivity of several polynomials in the Chern forms of a Griffiths
(semi)positive vector bundle not previously known, thus giving some new
evidences towards a conjecture by Griffiths, which in turn can be seen as a
pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on
numerically positive polynomials for ample vector bundles.
