Here is my question : I have a harmonic function $h$ on the open unit disc in $D \subset \mathbb{C}$, such that $\iint_D e^{2h} d\lambda(z) \leq A < \infty$ ($d\lambda$ is the Lebesgue measure on $\mathbb{C}$).

Can one have an upper bound of $e^h$ near the boundary of the disc ?

With elementary tools (mean value property for harmonic functions + Jensen's inequality with the exponential map) I can only obtain $e^{h(z)} \leq \frac{C^{te}}{d(z,S^1)}$ ($d(z,S^1)$ is the distance between $z$ and the boundary of $D$), but I would like something better (for example, $e^{h(z)} \leq \frac{C^{te}}{d(z,S^1)^{1/2}}$ or something like that would be nice).

If not, can one have an upper bound on the integral of $e^h$ over a ray passing through 0 ($\int_0^1 e^{h(r)} dr$ for example) ?

Thank you!