Lower bound on the smallest eigenvalue

Frobenius norm is the same as Euclidean norm and their squares is the sum of the squares of matrix entries. Therefore the bound you stated is wrong.

I think the bound you have encountered has the square of the maximum eigenvalue in the numerator instead of the Euclidean norm (Schindler's publication):http://library.utia.cas.cz/separaty/2009/AS/schindler-tikhonov%20regularization%20parameter%20in%20reproducing%20kernel%20hilbert%20spaces%20with%20respect%20to%20the%20sensitivity%20of%20the%20solution.pdf

But this bound is also wrong and should be corrected by the author. If you look into the proof you will notice preliminary mistakes.

If you're looking for a lower bound in terms of trace and determinant there are some publications available that you can trust. I can recommend this one for instance: https://www.researchgate.net/publication/242985986_Bounds_for_eigenvalues_using_the_trace_and_determinant