The Brennan conjecture is a mathematical hypothesis (in

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...

) for estimating (under specified conditions) the integral powers of the moduli of the derivatives of

conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...

s into the open unit disk. The conjecture was formulated by James E. Brennan in 1978. Let be a

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...

open subset of

\mathbb

with at least two boundary points in the

extended complex plane In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...

. Let

\varphi

be a conformal map of onto the open unit disk. The Brennan conjecture states that

\int_W , \varphi\ ', ^p\, \mathrmx\, \mathrmy < \infty

whenever

4/3 < p < 4

. Brennan proved the result when

4/3 < p < p_0

for some constant

p_0 > 3

. Bertilsson proved in 1999 that the result holds when

4/3 < p < 3.422

, but the full result remains open.

References

{{reflist Conjectures Unsolved problems in mathematics