Integrability of second derivative of conformal mappings a) How to construct a conformal mapping $f$ of the unit disk $D$ onto a Jordan domain with $C^1$ boundary such that $$\int_D|f''(z)|^2 dxdy =\infty.$$ (This is done in two different ways in the sequel)
b) It follows by Kellogg theorem that if $\partial D\in C^{1,\alpha}$ with $\alpha>1/2$, then $\int_D|f''(z)|^2 dxdy <\infty$ 
c) Motivated by b), I expect that if $\partial D\in C^{1,\alpha}$ with $0<\alpha\le 1/2$, then $\int_D|f''(z)|^2 dxdy <\infty$. 
 A: You should take a look at an example in a paper by Lesley and Warschawski published in Math Z.
The article is called "On conformal mappings with Derivative in VMOA" and is available at https://eudml.org/doc/172637.
A: Let $g$ be analytic in $\mathbb{D}$, continuous on $\overline{\mathbb{D}}$ such that $\int \int_{\mathbb{D}} |g'|^2=\infty$. It is well-known that such functions exist (the disc algebra is not contained in the Dirichlet space).
We can assume that $\operatorname{Re} g>0$ and $ | \operatorname{Im} g |<\pi/2$ on $\mathbb{D}$.
Then let $f$ such that $f'=e^g$. Since $|\arg{f'}|=|\operatorname{Im}g|<\pi/2$, we have that $\operatorname{Re}f'>0$ on $\mathbb{D}$, which is enough to conclude that $f$ maps $\mathbb{D}$ conformally onto some Jordan domain $G$ (see e.g. Pommerenke's book Boundary behaviour of conformal maps, Proposition 1.1).
Also, $\arg{f'}=\operatorname{Im}g$ is continuous on $\overline{\mathbb{D}}$, and so $\partial G$ is $C^1$ by Theorem 3.2 in Pommerenke's book.
Finally,
$\int \int_{\mathbb{D}} |f''|^2=\int \int_{\mathbb{D}}|g'e^g|^2 \geq \int \int_{\mathbb{D}}|g'|^2 = \infty,$
since $|e^g|=e^{\operatorname{Re}g}>1$.
A: The function $f(z)= z (\pi + 2 i \log z)$ maps a Jordan domain $\Omega$ bounded by a $C^2$ curve containing $(-1/2,1/2)$  onto a domain $\Omega'$ with $C^1$ boundary. Then $f''(z)=2/z$ so the given integral diverges. It can be composed by a conformal mapping of the unit disk onto $\Omega$ such that the given integral diverges as well. 
