  We consider a holomorphic family $f:\mathcal{X} \to S$ of compact complex
manifolds and a line bundle $\mathcal{L}\to \mathcal{X}$. Given that
$\mathcal{L}^{-1}$ carries a singular hermitian metric that has Poincar\'e type
singularities along a relative snc divisor $\mathcal{D}$, the direct image
$f_*(K_{\mathcal{X}/S}\otimes \mathcal{D} \otimes \mathcal{L})$ carries a
smooth hermitian metric. In case $\mathcal{L}$ is relatively positive, we give
an explicit formula for its curvature. The result applies to families of
log-canonically polarized pairs. Moreover we show that it improves the general
positivity result of Berndtsson-P\u{a}un in a special situation of a big line
bundle.
