  In this paper we compare a torsion free sheaf $\FF$ on $\PP^N$ and the free
vector bundle $\oplus_{i=1}^n\OPN(b_i)$ having same rank and splitting type. We
show that the first one has always "less" global sections, while it has a
higher second Chern class. In both cases bounds for the difference are found in
terms of the maximal free subsheaves of $\FF$. As a consequence we obtain a
direct, easy and more general proof of the "Horrocks' splitting criterion",
also holding for torsion free sheaves, and lower bounds for the Chern classes
$c_i(\FF(t))$ of twists of $\FF$, only depending on some numerical invariants
of $\FF$. Especially, we prove for rank $n$ torsion free sheaves on $\PP^N$,
whose splitting type has no gap (i.e. $b_i\geq b_{i+1}\geq b_i-1$ for every
$i=1, ...,n-1$), the following formula for the discriminant: \[
\Delta(\FF):=2nc_2-(n-1)c_1^2\geq -{1/12}n^2(n^2-1)\] Finally in the case of
rank $n$ reflexive sheaves we obtain polynomial upper bounds for the absolute
value of the higher Chern classes $c_3(\FF(t)), ..., c_n(\FF(t))$, for the
dimension of the cohomology modules $H^i\FF(t)$ and for the Castelnuovo-Mumford
regularity of $\FF$; these polynomial bounds only depend only on $c_1(\FF)$,
$c_2(\FF)$, the splitting type of $\FF$ and $t$.
