Brennan conjecture

In mathematics, specifically complex analysis, the Brennan conjecture is a conjecture estimating 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.
Let be a simply connected open subset of with at least two boundary points in the extended complex plane. Let be a conformal map of onto the open unit disk. The Brennan conjecture states that
whenever. Brennan proved the result when for some constant. Bertilsson proved in 1999 that the result holds when, but the full result remains open.