Brennan's conjecture

Theorem 1.1: for every $\varepsilon>0$ there is $C_\varepsilon$ with $M_{-2}[f'](r)\le C_\varepsilon(1-r)^{-1-\varepsilon}$ for all $f\in\mathcal S$ and $1/2\le r<1$; $B_{\mathcal S}(-2)=1$; and every conformal bijection $\varphi:W\to\mathbb D$ has $|\varphi'|^s$ area-integrable for $4/3<s<4$. Also $B_{\mathcal S}(t)=|t|-1$ for all $t\le-2$, and the Koebe function shows both endpoints fail. Not shown: the integral-means spectrum at other interior exponents; the companion's $B_b(-1)<1/4$ is a separate entry.

Theorem 1.1: for every $\varepsilon>0$ there is $C_\varepsilon$ with $M_{-2}[f'](r)\le C_\varepsilon(1-r)^{-1-\varepsilon}$ for all $f\in\mathcal S$ and $1/2\le r<1$; $B_{\mathcal S}(-2)=1$; and every conformal bijection $\varphi:W\to\mathbb D$ has $|\varphi'|^s$ area-integrable for $4/3<s<4$. Also $B_{\mathcal S}(t)=|t|-1$ for all $t\le-2$, and the Koebe function shows both endpoints fail. Not shown: the integral-means spectrum at other interior exponents; the companion's $B_b(-1)<1/4$ is a separate entry.