Some questions of uniqueness for extremal quasiconformal mappings

For each simply connected region Omega, we define a number h(Omega), 0 <= h(Omega) <= infinity, in terms of the family of holomorphic functions of class L-1 in Omega. It is known that the affine stretch of Omega is uniquely extremal if and only if h(Omega) = 0. We introduce a possible approach to estimating h(Omega) from below by solving a first-order partial differential equation, and illustrate it to give a new proof that h(Omega) = infinity for the case when Omega is a parabola.