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We prove$L$^{$q$}-inequalities for the gradient of the Green potential ($Gf$) in bounded, connected NTA-domains in$R$^{$n$},$n$≥2. These domains may have a highly non-rectifiable boundary and in the plane the set of all bounded simply connected NTA-domains coincides with the set of all quasidiscs. We get a restriction on the exponent$q$for which our inequalities are valid in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.

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.