Sharp bounds for eigenvalues of triangles

Abstract

We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane is bounded above by $\pi^2 L^2\over 9A^2$, where $L$ is the perimeter and $A$ is the area of this triangle. We show that the \mbox{constant 9} is optimal and that the optimal constant for the lower bound of the same form is 16. This gives a positive answer to a conjecture made by P. Freitas.