In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We illustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that
$ \lim_{p\to \infty}\lambda_{1,p}(\Omega) = \lambda_{1,\infty}(\Omega) = \left(\frac{\pi}{2R}\right)^2 $
where $ R $ is the largest radius of a ball included in the domain $ \Omega\subset {\mathbb R}^n $, and $ \lambda_{1,p}(\Omega) $ and $ \lambda_{1,\infty}(\Omega) $ are the principal eigenvalue for the homogeneous $ p $-laplacian and the homogeneous infinity laplacian respectively.
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