In this paper we compare a torsion free sheaf F on

and the free vector bundle

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 F. 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

(F(t)) of twists of F, only depending on some numerical invariants of F. Especially, we prove for rank n torsion free sheaves on

, whose splitting type has no gap (i.e.,

1 for every i = 1,

,n-1), the following formula for the discriminant:

. Finally in the case of rank n reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes

(F(t)),

,

(F(t)) for the dimension of the cohomology modules

and for the Castelnuovo-Mumford regularity of F; these polynomial bounds only depend only on

,

, the splitting type of F and t.