  Let $M$ and $N$ be two compact complex manifolds. We show that if the
tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and
$\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map
from $M$ to $N$. In particular, we prove that any holomorphic map from a
compact complex manifold $M$ with RC-positive tangent bundle to a compact
complex manifold $N$ with nef cotangent bundle must be a constant map. As an
application, we obtain that there is no non-constant holomorphic map from a
compact Hermitian manifold with positive holomorphic sectional curvature to a
Hermitian manifold with non-positive holomorphic bisectional curvature.
