Sharp bounds for eigenvalues of triangles
Bartłomiej A. Siudeja · The Michigan Mathematical Journal · 2007
We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane is bounded above by π²L²/9A², where L is the perimeter and A is the area of this triangle. We show that the 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 in [2].