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 L²(). Let w_ be the torsion function for , and let \|.\|ₚ denote the Lᵖ norm. It is shown that there exist {η₁>0,η₂>0} such that { (i) \|w_\|∞ λ()≥ 1+η₁ for any non-empty, open, simply connected set ⊂ ² with () >0, (ii) \|w_\|₁λ()≤ (1-η₂)|| for any non-empty, open, simply connected set ⊂² with finite measure ||.