Upper bounds for regularized determinants

Let $E$ be a holomorphic vector bundle on a compact K"ahler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $\mathrm{\Delta }$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $\mathrm{\Delta }$, one defines its regularized determinant $de{t}^{\mathrm{\prime }}(\mathrm{\Delta })$. We conjectured elsewhere that, when $h$ varies, this determinant $de{t}^{\mathrm{\prime }}(\mathrm{\Delta })$ remains bounded from above. In this paper we prove this in two special cases. The first case is when $X$ is a Riemann surface, $E$ is a line bundle and $dim(H^0 (X,E)) + dim(H^1 (X,E)) \leq 2$,andthesecondcaseiswhen$X$istheprojectiveline,$E$ is a line bundle, and all metrics under consideration are invariant under rotation around a fixed axis.