Monotonicity properties of the Robin torsion function in a class of symmetric planar domains

We prove the monotonicity property of the Robin torsion function in a smooth planar domain $\mathrm{\Omega }$ with a line of symmetry, provided that the Robin coefficient $\beta$ is greater than or equal to the negative of the boundary curvature $\kappa$ (i.e., $\beta \ge -\kappa$ on $\mathrm{\partial }\mathrm{\Omega }$). We also show that this condition is, in a certain sense, sharp by constructing a counterexample.