Let Mn\documentclass[12pt]{minimal}
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\begin{document}$$M^n$$\end{document} be a complete n-dimensional Riemannian manifold and Γf\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _f$$\end{document} the graph of a C2\documentclass[12pt]{minimal}
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\begin{document}$$C^2$$\end{document}-function f defined on a metric ball of Mn\documentclass[12pt]{minimal}
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\begin{document}$$M^n$$\end{document}. In the same spirit of the estimates obtained by Heinz for the mean and Gaussian curvatures of a surface in R3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^3$$\end{document} which is a graph over an open disk in the plane, we obtain in this work upper estimates for inf|R|\documentclass[12pt]{minimal}
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\begin{document}$$\inf |R|$$\end{document}, inf|A|\documentclass[12pt]{minimal}
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\begin{document}$$\inf |A|$$\end{document} and inf|Hk|\documentclass[12pt]{minimal}
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\begin{document}$$\inf |H_k|$$\end{document}, where R, |A| and Hk\documentclass[12pt]{minimal}
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\begin{document}$$H_k$$\end{document} are, respectively, the scalar curvature, the norm of the second fundamental form and the k-th mean curvature of Γf\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _f$$\end{document}. From our estimates we obtain several results for graphs over complete manifolds. For example, we prove that if Mn,n≥3,\documentclass[12pt]{minimal}
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\begin{document}$$M^n,\;n\ge 3,$$\end{document} is a complete noncompact Riemannian manifold with sectional curvature bounded below by a constant c, and Γf\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _f$$\end{document} is a graph over M with Ricci curvature less than c, then inf|A|≤3(n-2)-c\documentclass[12pt]{minimal}
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\begin{document}$$\inf |A|\le 3(n-2)\sqrt{-c}$$\end{document}. This result generalizes and improves a theorem of Chern for entire graphs in Rn+1\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb R^{n+1}$$\end{document}.
