I'm getting stuck with a proposition. Please can someone help me? Let $E$ be a holomorphic vector bundle with hermitian metric $h$ over a connected compact Kähler manifold $X$ with Kähler form $\omega$ and associated Kähler metric $g$. Let $s\in H^{0}(X,E)$ a nonzero section. Consider the current $T_{s}=\frac{i\partial\bar{\partial}log(h(s,s))}{2\pi}$. Let's write $F(h)$ for the curvature of the Chern connection respect to the metric $h$, and another current $R_s=\frac{ih(F(h)s,s)}{2\pi h(s,s)}$. How can I prove that $0 \leq T_{s}+R_{s}$ in the following sense: (regarding those currents as forms) the hermitian form associated to $T_{s}+R_{s}$ is semipositive definite. The Hermitian form associated to a (1,1)real form $\delta$ is the one given in local coordinates by the matrix of coefficients of $i\delta$ and is (semi)positive if this matrix is.

This is a consequence of the generalized LelongPoincaré formula for vector bundles: denoting $D'$ the $(1,0)$ part of the Chern connection of $(E,h)$ and $\Theta_h(E)$ its Chern curvature, one has: $$dd^c \log s^2=\frac{1}{s^2}\cdotp \left( D's^2\frac{\langle D's,s\rangle^2}{s^2} \langle \Theta_h(E)s, s\rangle\right)$$ and essentially by CauchySchwartz inequality, one has $$s^2\cdotp D's^2\langle D's,s\rangle^2 \ge 0.$$ 

