  We show that if $E$ is an ample vector bundle of rank at least two with some
curvature bound on $O_{P(E^*)}(1)$, then $E^*\otimes \det E$ is Kobayashi
positive. The proof relies on comparing the curvature of $(\det E^*)^k$ and
$S^kE$ for large $k$ and using duality of convex Finsler metrics. Following the
same thread of thought, we show if $E$ is ample with similar curvature bounds
on $O_{P(E^*)}(1)$ and $O_{P(E\otimes \det E^*)}(1)$, then $E$ is Kobayashi
positive. With additional assumptions, we can furthermore show that $E^*\otimes
\det E$ and $E$ are Griffiths positive.
