I have the following question:

Let $u$ be a smooth subharmonic function on the unit disc $\mathbb{D}:=\left\{ z\in\mathbb{C}:\left|z\right|<1\right\} $. Assume that $u=0$ on the boundary of $\mathbb{D}$ and $$ \int_{\mathbb{D}}\Delta u=1. $$ Here $\Delta u$ is the Riesz measure of $u$.

Is there any chance to show that $$ \Delta u\leq\frac{M}{1-\left|z\right|^{2}}dV $$ as measures, for some positive constant $M$ (possibly an absolute constant, i.e. independent of $u$)?

Here $dV$ is the standard Lebesgue measure on $\mathbb{D}$.

Thanks for any suggestions.