5
$\begingroup$

Let $f$ be a conformal mapping of the unit disk $U$ into $C$. Is the following integral convergent $$\int_U \frac{dx dy}{|f'(z)|}?$$

$\endgroup$
4
$\begingroup$

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 $\delta.$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.