A sharp two-sided eigenvalue bound for triangles

Let $T$ be a triangle in the plane with area $A$ and perimeter $L$, and let $\lambda_T$ denote the first eigenvalue of the Dirichlet Laplacian on $T$.

Triangle eigenvalue bound. The eigenvalue satisfies

$$\frac{\pi^2L^2}{16A^2}+\frac{7\sqrt{3}\pi^2}{12A}\leq \lambda_T\leq \frac{\pi^2L^2}{12A^2}+\frac{\sqrt{3}\pi^2}{3A}.$$

The lower bound is sharp asymptotically for tall isosceles triangles, so its constant cannot be decreased; the upper bound was previously conjectured and supported by numerical evidence. The upper bound is attained by equilateral triangles, while its validity in general remains open in the source.