$f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|\leq 2\}$ such that $\iint_D=|f(z)|^2\,dx\,dy\leq 3\pi$. Maximize $|f''(0)|$ 
Determine the largest possible value of $|f''(0)|$ when $f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|<2\}$ with the property that $\iint_{D}|f(z)|^2\,dx\,dy\leq 3\pi$.

I don't really know what to do with the assumption that $\iint_D|f(z)|^2\,dx\,dy\leq 3\pi$. I believe you could use Stokes' theorem to rewrite this as a line integral on $\partial D$, but I'm really rusty with my usage, so I'm kind of stuck.
If I could get a bound on $\int_{\partial D}|f(z)|^2\,dz$, I could probably use harnack's inequality for subharmonic functions to get a bound on $|f|$ then Cauchy's inequality, however I'm not very sure about my usage of green's theorem (if that's even the right way to go) any help is greatly appreciated. Thanks

Hint: Write $f$ as a power series, use polar coordinates on $\int_D |f|^2,$ and use the orthogonality of the exponentials.
