Brennan conjecture explained

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]}

Let be a simply connected open subset of

C with at least two boundary points in the extended complex plane. Let \varphi be a conformal map of onto the open unit disk. The Brennan conjecture states that\int _{W} |\varphi '| ^{p} d x d y < inf ty whenever 4 / 3 < p < 4 . Brennan proved the result when 4 / 3 < p < p _{0} for some constant p _{0} > 3 .^[1] Bertilsson proved in 1999 that the result holds when 4 / 3 < p < 3 . 422 , but the full result remains open.^{[3] [4]}

Notes and References

1. Brennan, James E.. The integrability of the derivative in conformal mapping. Journal of the London Mathematical Society. 2. 2. 1978. 261–272. 10.1112/jlms/s2-18.2.261.
2. Web site: A brief review on Brennan's conjecture, Malaga, July 10–14, 2011. Stylogiannis, Georgios (Aristotle University of Thessaloniki, Greece).
3. Hu. J.. Chen, S.. A better lower bound estimation of Brennan's conjecture. 2015. 1509.00270. math.CV.
4. Book: Bertilsson, Daniel. On Brennan's conjecture in conformal mapping. Kungliga Tekniska Högskolan. 1999.

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