Brennan's conjecture and sharp inverse-square integral means

We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then $|{\phi }^{\prime }{|}^s$ is area-integrable for every $4/3<s<4$. We also prove the sharp inverse-square integral-means exponent $B_{\mathcal{S}}(-2)=1$ for the normalized schlicht class $\mathcal{S}$.