Lemma (Arithmetic-Geometric Mean Inequality)

For any sequence of $k$ nonnegative real numbers $b_{1},\dots,b_{k}$ $$\left( \prod_{i=1}^{k}b_{i} \right)^{\frac{1}{k}} \le \frac{1}{k}\sum_{i=1}^{k}b_{i}$$where equality holds if and only if $b_{i}=b, i=1,\dots,k$.

Lemma (Hadamard’s Inequality)

Let $\mathbf{R}$ be a positive semidefinite matrix with diagonal elements $r_{ii}, i=1,\dots,k$. Then $$\det \mathbf{R}\le \prod_{i=1}^{k}r_{ii}$$If $\mathbf{R}$ is positive definite, then equality holds if and only if $\mathbf{R}$ is diagonal.