  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 $\Delta$ acting upon
smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one
defines its regularized determinant $det'(\Delta)$. We conjectured elsewhere
that, when $h$ varies, this determinant $det'(\Delta)$ 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 $dim(H^0 (X,E)) + dim(H^1 (X,E))
\leq 2$, and the second case is when $X$ is the projective line, $E$ is a line
bundle, and all metrics under consideration are invariant under rotation around
a fixed axis.
