Your second inequality is a consequence  of the Hadamard inequality, see here:
http://en.wikipedia.org/wiki/Hadamard_inequality

Another way is to use the result that the eigenvalue vector of a Hermitian (hence real symmetric) matrix majorizes the vector of diagonal values, and then using that the product function $\prod_{i=1}^n x_i$ for $x_i&gt;0$ is Schur-convex:

One source for this results is the book by Bhatia: "Matrix Analysis". 

Upper Bound on Determinant of Matrix in terms of Trace
I have been wondering something for a while now. We have a manual penalty from Google on unnatural linking. When I check out "Links to your site" in Webmaster Tools we have 11,387 links from Meetup.com. Is this harming us? After clicking through some of the links randomly I have discovered it is from one of our dealers that sells our products. He is sponsoring different groups and so the ad appears, the problem is he is linking back to our site and not his own. The anchor text is all the same it never changes. 
