A new proof of the bound for the first Dirichlet eigenvalue of the Laplacian operator

In this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue λ1D(B(p,r))$\lambda _1^D \left({B\left({p,r} \right)} \right)$ of Laplacian operator for the manifold with Ricci curvature Rc ≥ −K, by using Li-Yau’s gradient estimate for the heat equation.