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 $Ω$ with a line of symmetry, provided that the Robin coefficient $β$ is greater than or equal to the negative of the boundary curvature $κ$, i.e., $β\geq - κ$ on $\partialΩ$. We also construct a counterexample