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This work concerns bounds for Chern classes of holomorphic semistable and stable vector bundles on ${\mathbb{P}}^n$. Non-negative polynomials in Chern classes are constructed for 4-vector bundles on ${\mathbb{P}}^4$ and a generalization of the presented method to r-bundles on ${\mathbb{P}}^n$ is given. At the end of this paper the construction of bundles from complete intersection is introduced to see how rough the estimates we obtain are.