Brennan conjecture

The Brennan conjecture asserts that every conformal map from the unit disk onto a simply connected domain has a finite area integral of the pth power of its derivative for −2/3 < p < 2.

Connect

Conformal map: A holomorphic function that preserves angles wherever its derivative is nonzero. The conjecture quantifies the derivative of maps in this class.
Brennan's conjecture paper: A 1978 paper by James E. Brennan that formulated the derivative-integrability conjecture. It introduced the specific range of exponents now associated with the conjecture.
Brennan conjecture for negative exponents: The part of Brennan's conjecture asserting derivative-power integrability for −2/3 < p < 0. Negative powers test how often the conformal-map derivative becomes small.
Quasiconformal mapping: A homeomorphism with weak derivatives that distorts infinitesimal circles by a uniformly bounded ratio. Quasiconformal methods provide extensions and estimates used in conformal-map regularity.
Bieberbach conjecture: The conjecture that the nth Taylor coefficient of a normalized univalent function has modulus at most n. It is another famous univalent-function problem, but concerns coefficients rather than derivative integrals.
Unit disk: The set of complex numbers whose modulus is less than one. It is the fixed source domain for the conformal maps in the conjecture.
James E. Brennan: An American mathematician known for work in complex analysis, especially conformal mapping. He formulated the conjecture while studying integrability of conformal-map derivatives.
Brennan conjecture for positive exponents: The part of Brennan's conjecture asserting derivative-power integrability for 0 < p < 2. Positive powers test the growth of the derivative near boundary singularities.
Beurling transform: A singular integral operator central to the analysis of planar quasiconformal maps. Its integrability estimates connect quasiconformal distortion with derivative bounds.
Makarov's theorem: A result bounding the dimension of harmonic measure on simply connected planar domains. It gives broad boundary information without establishing Brennan's full integral estimate.