I am looking at problem 14.5 on the book Heat Kernel and Analysis on Manifolds by Grigor’yan. The problem goes like this, if a weighted manifold $(M,g,\mu)$ satisfies the following Faber-Krahn inequality $$\lambda_1(\Omega)\geq a\mu(\Omega)^{-\frac{2}{\nu}}$$ with some $a>0$, $\nu>0$ for any relatively compact set $\Omega\subseteq M$, where $\lambda_1$ is the first eigenvalue of the Laplacian operator on $\Omega$ with Dirichlet boundary condition. Then this implies that for any relatively compact ball $B(x,r)\subseteq M$ that $$\mu(B(x,r))\geq ca^{\nu/2}r^{\nu}$$ where $c=c(\nu)$. The hint says first prove that $$\mu(B(x,r))\ge c(ar^2)^{\frac{\nu}{\nu+2}}\mu(B(x,r))^{\frac{\nu}{\nu+2}}$$ then iterate this inequality. The iteration process makes sense, but I have no clue how to derive the above relative estimate on $\mu(B(x,r))$.
Any comments and ideas are appreciated, thanks in advance.