  Motivated by Koll\'{a}r-Matsusaka's Riemann-Roch type inequalities, applying
effective very ampleness of adjoint bundles on Fujita conjecture we show that
for any partition $\lambda$ of the positive integer $d$ there exists a
universal bivariate polynomial $Q_\lambda(x, y)$ which has $\text{deg}\ Q \leq
d$ and whose coefficients depend only on $n$, such that for any projective
manifold $X$ of dimension $n$ and any ample line bundle $L$ on $X$,
\begin{equation*}
  \left|c_\lambda(X)\cdot L^{n -d}\right|\leq
  \frac{Q_{\lambda}(L^{n}, K_X \cdot L^{n -1} )}{(L^{n})^{d-1}},
\end{equation*} where $K_X$ is the canonical bundle of $X$ and $c_\lambda(X)$
is the monomial Chern class given by the partition $\lambda$. As a special
case, when $K_X$ or $-K_X$ is ample, this implies that there exists a constant
$c_n$ depending only on $n$ such that for any monomial Chern classes of top
degree, the Chern number ratios \begin{equation*} \left|\frac{c_\lambda(X)}{c_1
(X) ^{n}}\right|\leq c_n, \end{equation*} which recovers a recent result of
Du-Sun. The main result also yields an asymptotic version of the sharper
Riemann-Roch type inequality.

Algebraic Geometry;Differential Geometry