Kraetzer's conjecture on the universal integral means spectrum

Theorem 1.1: there are constants $0<\varepsilon<1/4$ and $C$ with $M_{-1}[f'](r)\le C(1-r)^{-1/4+\varepsilon}$ for every $f\in\mathcal S$ and $1/2\le r<1$, so $B_b(-1)<1/4=p^2/4$ at $p=-1$ and Kraetzer's formula is false. The bound needs no bounded image or boundary regularity. Not shown: the true value of $B_b(-1)$ or any explicit gap, and nothing about $p=1$ or the Carleson-Jones question; Brennan's value $B(-2)=1$, also predicted by Kraetzer, is proved in the companion (separate entry).

Theorem 1.1: there are constants $0<\varepsilon<1/4$ and $C$ with $M_{-1}[f'](r)\le C(1-r)^{-1/4+\varepsilon}$ for every $f\in\mathcal S$ and $1/2\le r<1$, so $B_b(-1)<1/4=p^2/4$ at $p=-1$ and Kraetzer's formula is false. The bound needs no bounded image or boundary regularity. Not shown: the true value of $B_b(-1)$ or any explicit gap, and nothing about $p=1$ or the Carleson-Jones question; Brennan's value $B(-2)=1$, also predicted by Kraetzer, is proved in the companion (separate entry).