  Let $E$ be a holomorphic vector bundle over a compact K\"{a}hler manifold
$(X,\omega)$ with negative sectional curvature $sec\leq -K<0$, $\Delta_{E}$ be
the Chern connection on $E$. In this article we show that if
$C:=|[\Lambda,i\Theta(E)]|\leq c_{n}K$, then $(X,E)$ satisfy a family of Chern
number inequalities. The main idea in our proof is study the $L^{2}$
$\bar{\partial}_{\tilde{E}}$-harmonic forms on lifting bundle $\tilde{E}$ over
the universal covering space $\tilde{X}$. We also observe that there is a
closely relationship between the eigenvalue of the Laplace-Beltrami operator
$\Delta_{\bar{\partial}_{\tilde{E}}}$ and the Euler characteristic of $X$.
Precisely, if there is a line bundle $L$ on $X$ such that
$\chi^{p}(X,L^{\otimes m})$ is not constant for some integers $p\in[0,n]$, then
the Euler characteristic of $X$ satisfies $(-1)^{n}\chi(X)\geq
(n+1)+\lfloor\frac{c_{n}K}{2nC} \rfloor$.
