Let$f$be a one-to-one analytic function in the unit disc with$f′$(0)=1. We prove sharp estimates for certain Taylor coefficients of the functions$(f′)$^{$p$}, where$p$<0. The proof is similar to de Branges’ proof of Bieberbach’s conjecture, and thus relies on Löwner’s equation. A special case leads to a generalization of the usual estimate for the Schwarzian derivative of$f$. We use this to improve known estimates for integral means of the functions |$f′$|^{$p$}for integers$p$⪯−2.