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 so-called 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 $K(z)=z/(1-z)^2$ 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 $\delta.$


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