Let $u$ be a function on a domain $\Omega\subset R^n$, and $D^2u=\left(\frac{\partial^2 u}{\partial x_i\partial x_j}\right)_{n\times n}$ be the Hessian of $u$. If $D^2u$ is positively definite and $\det(D^2u)<\mu$ for some constant $\mu$, then $$ |D^2u|\leq (C(\varepsilon)+\varepsilon M)\sum_{i=1}^nu^{ii}, $$ holds for any $\varepsilon>0$, where $C(\varepsilon)$ is a constant depending only on $\varepsilon$, $(u^{ij})_{n\times n}$ is the inverse of $D^2u$, and $M=\sup_{x\in\Omega}|D^2u|$ .
I have tried the prove that in the case of $D^2u=\mathrm{diag}\{\lambda_1,\lambda_2,\ldots,\lambda_n\}$, but stacked for $n=3$. (in this case, n=1 and 2 are trivial.)