Positivity of direct images with a Poincar\'e type twist

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 $f_{\ast }(K_{\mathcal{X}\mathrm{/}S}\otimes \mathcal{D}\otimes \mathcal{L})$ 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.