On the torsion function for simply connected, open sets in ^2

For an open set ⊂^2 let λ() denote the bottom of the spectrum of the Dirichlet Laplacian acting in L^2(). Let w_ be the torsion function for , and let ._p denote the L^p norm. It is shown there exists η>0 such that w__∞λ()≥ 1+η for any non-empty, open, simply connected set ⊂^2 with () >0. Moreover, if the measure || of is finite, then w__1λ()≤ (1-η)||.