Let $f$ be a conformal mapping of the unit disk $U$ into $C$. Is the following integral convergent $$\int_U \frac{dx dy}{f'(z)}?$$
Problems of this type are part of the socalled Brennan's conjecture. More precisely, suppose that $f:\mathbb{D} \to \mathbb{C}$ is univalent. Brennan's conjecture states that $$\int_{\mathbb{D}}f'^p dA < \infty$$ for $2<p<2/3$. I am not an expert on this subject, but apparently it is known since the work of Shimurin (2005) that Brennan's conjecture is true for $1.78 < p < 2/3$. EDIT The Koebe function $K(z)=z/(1z)^2$ shows that the above integral diverges for $p \leq 2$ and for $p \geq 2/3$. Moreover, an elementary calculation involving Koebe's distortion theorem and a theorem of Pravitz shows that Brennan's Conjecture holds for $1<p<2/3$. Brennan himself (see The integrability of the derivative in conformal mapping. J. London Math. Soc. (2) 18 (1978)) showed that the Conjecture holds for $1\delta < p<2/3$ for some positive $\delta.$ 

