Brennan conjecture

The Brennan conjecture is a mathematical hypothesis (in complex analysis) for estimating (under specified conditions) the integral powers of the moduli of the derivatives of conformal maps into the open unit disk. The conjecture was formulated by James E. Brennan in 1978.^[1][2][3]

Let W be a simply connected open subset of ${\displaystyle \mathbb {C} }$ with at least two boundary points in the extended complex plane. Let ${\displaystyle \varphi }$ be a conformal map of W onto the open unit disk. The Brennan conjecture states that ${\displaystyle \int _{W}|\varphi \ '|^{p}\,\mathrm {d} x\,\mathrm {d} y<\infty }$ whenever ${\displaystyle 4/3<p<4}$. Brennan proved the result when ${\displaystyle 4/3<p<p_{0}}$ for some constant ${\displaystyle p_{0}>3}$.^[1] Bertilsson proved in 1999 that the result holds when ${\displaystyle 4/3<p<3.422}$, but the full result remains open.^[4][5]

References

1. Brennan, James E. (1978). "The integrability of the derivative in conformal mapping". Journal of the London Mathematical Society. 2 (2): 261–272. doi:10.1112/jlms/s2-18.2.261.
2. James E. Brennan at the Mathematics Genealogy Project
3. Stylogiannis, Georgios (Aristotle University of Thessaloniki, Greece). "A brief review on Brennan's conjecture, Malaga, July 10–14, 2011" (PDF).{{cite web}}: CS1 maint: multiple names: authors list (link)
4. Hu. J.; Chen, S. (2015). "A better lower bound estimation of Brennan's conjecture". arXiv:1509.00270 [math.CV].
5. Bertilsson, Daniel (1999). On Brennan's conjecture in conformal mapping (PDF). Kungliga Tekniska Högskolan; 110 pages{{cite book}}: CS1 maint: postscript (link)