  By using analytic method, we prove that there exist rational curves on
compact Hermitian manifolds with positive holomorphic bisectional curvature. It
confirms a question of S.-T. Yau. It is well-known that Mori proved in
\cite{Mori79} that every compact complex manifold $N$ with $c_1(N)>0$ contains
at least one rational curve. However, as a borderline example, we show that the
standard Hopf surface $S^1\times S^3$ has a Hermitian metric with non-negative
holomorphic bisectional curvature (in particular, $c_1(S^1\times S^3)\geq 0$),
but it contains no rational curve.
