Heating of the Ahlfors-Beurling operator, and estimates of its norm

A new estimate is established for the norm of the Ahlfors-Beurling transform $T\varphi(z):=\frac1\pi\iint\frac{\varphi(\zeta)\, dA(\zeta)}{(\zeta - z)^2}$ in $L^p(dA)$. Namely, it is proved that $\Vert T\Vert _{L^p\rightarrow L^p} \leq 2(p-1)$ for all $p\geq 2$. The method of Bellman function is used; however, the exact Bellman function of the problem has not been found. Instead, a certain approximation to the Bellman function is employed, which leads to the factor 2 on the right (in place of the conjectural $1$).