I'd like to suggest another approach to the problem. Consider this theorem of Bai & Golub (quoted as 2.1 in the paper http://gerard.meurant.pagesperso-orange.fr/trace_2009.pdf):

Theorem If $A$ is a symmetric real positive definite $n \times n$ matrix whose eigenvalues lie in $[a,b]$, $\mu_{1}=tr(A),\mu_{a}=||A||^{2}_{F}$, then:

$ \begin{bmatrix} \mu_{1} & n \end{bmatrix} \begin{bmatrix}\mu_{2} & \mu_{1} \\\\ b^{2} & b\end{bmatrix}^{-1} \begin{bmatrix} n \\\\ 1 \end{bmatrix} \leq Tr(A^{-1})$

I think you can use it together with Weyl's theorem to good effect.

I'd like to suggest another approach to the problem. Consider this theorem of Bai & Golub (quoted as 2.1 in the paper http://gerard.meurant.pagesperso-orange.fr/trace_2009.pdf):

Theorem If $A$ is a symmetric real positive definite $n \times n$ matrix whose eigenvalues lie in $[a,b]$, $\mu_{1}=tr(A),\mu_{2}=||A||^{2}_{F}$, then:

$ \begin{bmatrix} \mu_{1} & n \end{bmatrix} \begin{bmatrix}\mu_{2} & \mu_{1} \\\\ b^{2} & b\end{bmatrix}^{-1} \begin{bmatrix} n \\\\ 1 \end{bmatrix} \leq Tr(A^{-1})$

I think you can use it together with Weyl's theorem to good effect.