The first eigenvalue of a scalene triangle

Abstract

In this paper, we prove the lower bound \[ λ 1 ( T ) ≥ ( L + 4 π A ) 2 16 A 2 {\lambda _1}(T) \geq \frac {{(L + \sqrt {4\pi A{)^2}} }}{{16{A^2}}} \] for a triangle T T with area A A and perimeter L L , where λ 1 {\lambda _1} is the first eigenvalue for the Laplace operator with Dirichlet boundary conditions. We also present analogous estimates for an arbitrary convex polygon.