On the torsion function for simply connected, open sets in ${\mathbb{R}}^2$

For an open set $\text{Om}\subset {\mathbb{R}}^2$ let $\lambda (\text{Om})$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\text{Om})$. Let $w_{\text{Om}}$ be the torsion function for $\text{Om}$, and let $\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_p$ denote the $L^p$ norm. It is shown that there exist {${\eta }_1>0,{\eta }_2>0$} such that { (i) $\mathrm{\parallel }{w}_{\text{Om}}{\mathrm{\parallel }}_{\mathrm{\infty }}\lambda (\text{Om})\ge 1+{\eta }_1$ for any non-empty, open, simply connected set $\text{Om}\subset {\mathbb{R}}^2$ with $\text{lb}(\text{Om})>0$, (ii) $\mathrm{\parallel }{w}_{\text{Om}}{\mathrm{\parallel }}_1\lambda (\text{Om})\le (1-{\eta }_2)\mathrm{\mid }\text{Om}\mathrm{\mid }$ for any non-empty, open, simply connected set $\text{Om}\subset {\mathbb{R}}^2$ with finite measure $\mathrm{\mid }\text{Om}\mathrm{\mid }$.