Burkholder integrals, Morrey's problem and quasiconformal mappings

Inspired by Morrey's Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals $\text {B}_p$, $p \geqslant 2$, are quasiconcave, when tested on deformations of the identity $f\in \textup {Id} + {\mathscr C}^\infty _\circ (\Omega )$ with $\text {B}_p\,(Df(x)) \geqslant 0$ pointwise, or equivalently, deformations such that $\vert D f \vert^2 \leqslant \frac {p}{p-2}J_f$. In particular, quasiconcavity holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible $\mathscr L^p$-estimates for the gradient of a principal solution to the Beltrami equation $f_{\bar {z}}=\mu (z) f_z$, for any $p$ in the critical interval $2\leqslant p \leqslant 1+1/\Vert\mu \Vert _\infty$.