Trace-Determinant Inequality for MLE Estimation of Multivariate Normal Distribution

Question

I have the following statement in my lecture notes and cannot see why it is true. For given positive definite $p\times p$ matrices $\Sigma$, $B$ and a scalar $b>0$ we have $$ \frac{1}{\det(\Sigma)^p}e^{-tr(\Sigma^{-1}B)}\leq \frac{1}{\det(B)^p}(2b)^{pb}e^{-bp}. $$ It occurs in the context of finding the maximum likelihood estimator of a multivariate normal distribution. Are there any well known inequalities I can derive it from?

Answer 1

Taking $\Sigma = p^{-1} I_p$ and $B =I_p$ then the LHS is $e^{-p^2} p^{p^2} = (e^{-p} p^p)^p$. Note that $e^{-p} p^p \to \infty$ as $p\to \infty$ while $\inf_{b >0} (2b)^b e^{-b} = e^{-1/2}$. Then the inequality can not hold.