Some conjectures about integral means of $\mathrm{\partial }f$ and $\bar{\mathrm{\partial }}f$

We discuss some conjectural inequalities that are related to singular integrals, martingales, quasiconformal mappings, and the calculus of variations. Specifically, we present evidence for a conjecture of Iwaniec concerning the best constant for the Beurling-Ahlfors Operator.