Heinz type estimates for graphs in Euclidean space

Let $M^n$ be an entire graph in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$. Denote by $H$, $R$ and $\vert A\vert$, respectively, the mean curvature, the scalar curvature and the length of the second fundamental form of $M^n$. We prove that if the mean curvature $H$ of $M^n$ is bounded, then $\inf_M\vert R\vert=0$, improving results of Elbert and Hasanis-Vlachos. We also prove that if the Ricci curvature of $M^n$ is negative, then $\inf_M\vert A\vert=0$. The latter improves a result of Chern as well as gives a partial answer to a question raised by Smith-Xavier. Our technique is to estimate $\inf\vert H\vert,\;\inf\vert R\vert$ and $\inf\vert A\vert$ for graphs in $\mathbb{R}^{n+1}$ of $C^2$ real-valued functions defined on closed balls in $\mathbb{R}^n$.