The inradius, the first eigenvalue, and the torsional rigidity of curvilinear polygons

Let λ₁, P, and ρ denote the first eigenvalue of the Dirichlet Laplacian, the torsional rigidity, and the inradius of a planar domain Ω, respectively. In this paper, we prove several inequalities for λ₁, P, and ρ in the case when Ω is a curvilinear polygon with n sides, each of which is a smooth arc of curvature at most κ. Our main proofs rely on the method of dissymmetrization and on a special geometrical 'containment theorem' for curvilinear polygons. For rectilinear n-gons, which constitute a proper subclass of curvilinear n-gons with curvature at most 0, these inequalities were established by the first author in 1992. In the simplest particular case of Euclidean triangles T, the inequality linking the first eigenvalue and the inradius of T is equivalent to the inequality λ₁ ≤ π² L²/9 A ², where A and L denote the area and perimeter of T, respectively, which was recently discussed by Freitas (Proc. Amer. Math. Soc. 134 (2006) 2083-2089) and Siudeja (Michigan Math. J. 55 (2007) 243-254).